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arXiv:2210.08558v1 [math.RT] 16 Oct 2022
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES
AND ABELIAN MODEL STRUCTURES
ZHENXING DI, LIPING LI, LI LIANG, AND NINA YU
Abstract. In this paper we systematically consider representations over diagrams of abelian cat-
egories, which unify quite a few notions appearing widely in literature such as representations of
categories, sheaves of modules over categories equipped with Grothendieck topologies, representa-
tions of species, etc. Since a diagram of abelian categories is a family of abelian categories glued by
an index category, the central theme of our work is to determine whether local properties shared by
each abelian category can be amalgamated to the corresponding global properties of the represen-
tation category. Specifically, we investigate the structure of the representation categories, describe
important functors and adjunction relations between them, and construct cotorsion pairs in the rep-
resentation category by local cotorsion pairs in each abelian category. As applications, we establish
abelian model structures on the representation category for some particular index categories, and
characterize special homological objects (such as projective, injective, flat, Gorenstein injective, and
Gorenstein flat objects) in categories of presheaves of modules over some combinatorial categories.
Contents
Introduction 2
Part I. Diagrams and their representations 11
1. Diagrams of categories 11
2. Representations over diagrams of categories 15
2.1. Left representations over diagrams of categories 15
2.2. Limits and colimits 18
2.3. Right representations over diagrams of categories 20
3. Functors induced by morphisms between diagrams 21
4. Representations over diagrams of module categories 24
4.1. Diagrams of module categories 24
4.2. Representations over diagrams of module categories 26
Part II. Functors, adjunctions, and their applications 28
5. The induction functor and its applications 28
5.1. The restriction functor and its left adjoint 28
5.2. Grothendieck structure and locally finitely presented property 34
5.3. Dual results for Rep-D36
6. The lift and stalk functors 38
6.1. The lift functor and its left adoint 38
6.2. The stalk functor and its left adjoint 44
6.3. Dual results for Rep-D45
Date: October 18, 2022.
2010 Mathematics Subject Classification. 18G25; 18A25; 18A40.
Key words and phrases. Diagrams of categories; representations over diagrams of categories; (Gorenstein) homo-
logical representations; rooted categories; cotorsion pairs; abelian model structures.
1
2 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
7. Rooted categories and characterizations of projectives and injectives 46
7.1. Rooted categories 46
7.2. Characterizations of projectives and injectives 49
Part III. Abelian model structures and homological objects 55
8. Preliminaries on cotorsion pairs and abelian model structures 55
9. Induced cotorsion pairs in D-Rep 57
10. Abelian model structures on D-Rep 62
10.1. Completeness of the induced cotorsion pairs 62
10.2. Abelian model structures on D-Rep 66
11. Applications to diagrams of module categories 68
11.1. Flat objects 69
11.2. Gorenstein injective model structures 70
11.3. Gorenstein flat model structures 74
Appendix A: Diagrams of small preadditive categories 79
Appendix B: Details in proofs 85
Acknowledgments 91
References 92
Introduction
Motivation. Let (X, OX) be an arbitrary scheme, and let Pbe the poset of all affine open subsets
in Xordered by inclusions. By the descent property of quasi-coherent OX-modules, one obtains
an equivalence between the category of quasi-coherent sheaves on Xand the category of Cartesian
OX-modules on P, which are defined in [25] and [54] as follows: an R-module M, where Ris a
presheaf of commutative rings over P, is Cartesian if for any morphism α:i→jin P, the R(i)-
module homomorphism R(i)⊗R(j)M(j)→M(i) via ri⊗mj7→ riM(αop )(mj) is an isomorphism.
If further Ris flat (that is, R(αop) : R(j)→R(i) is a flat ring homomorphism for any morphism
α:i→jin P), then the category of Cartesian R-modules is Grothendieck; see [25, Proposition 4.2].
One can interpret the above construction in the framework of 2-categories. Explicitly, the small
category Pcan be viewed as a 2-category in a natural way and presheaves of commutative rings
over Pcan be viewed as contravariant 2-functors. This is a very special case of the notion of
I-diagrams of categories with Ia small category (see Definition 1.1), which appears widely in the
literature under different names. For instance, they are called I-indexed categories by Johnstone
[45]. Also, the pseudo lax functors from Ito the 2-category Cat of small categories, described by
Street in [59], are exactly I-diagrams of small categories. Given an I-diagram Dof categories,
there is a coCartesian fibration from the Grothendieck construction RIDto I. The Grothendieck
construction provides a classical correspondence between diagrams of categories and coCartesian
fibrations over the index category. This machinery was first applied to diagrams of sets by Yoneda
and later developed in full generality by Grothendieck in [37]; see also [49, I.5]. For basic facts and
homological properties on Grothendieck constructions of diagrams of categories, one can refer to
e.g. [2, 6, 38, 60].
The notion of representations over diagrams of categories was given by Mozgovoy in [51], which
are also called twisted representations by Gothen and King in [34] or twisted diagrams by H¨uttemann
and R¨ondigs in [44]. Explicitly, given an I-diagram Dof categories, a left representation Mover
Dis a rule to assign
•an object Miin Dito any i∈Ob(I), and
•a morphism Mα:Dα(Mi)→Mjto any α:i→jin Mor(I)
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 3
such that two axioms are satisfied; see Definition 2.1 for details. Denote by D-Rep the category
of all left representations over D. Mozgovoy pointed out in [51, Remark 2.7] that the category of
sections of the associated coCartesian fibration RID→Iis equivalent to D-Rep.
We notice that D-Rep unifies many special cases such as comma categories, module categories
of Morita context rings, categories of additive functors from Ito an abelian category (which are
called representations of Iby representation theorists), categories of representations of (generalised)
species and phyla studied in e.g. [15, 16, 27, 28, 29, 46, 47]; see Examples 2.2 and 2.4. Another
example comes from a recent work by Estrada and Virili [26]. Roughly speaking, given an I-diagram
Rof associative rings1, we can construct an I-diagram Rof module categories with
•Ri=Ri-Mod, the category of left Ri-modules for any i∈Ob(I) and
•Rα=Rj⊗Ri−for any α:i→jin Mor(I)
such that R-Rep coincides with the category R-Mod of left R-modules in the sense of Estrada and
Virili [26]2, which is also the category of presheaves of modules over the structure presheaf Rof
rings with respect to the opposite category Iop ; see Subsection 4.1 and Proposition 4.5.
As we mentioned above, diagrams of categories and representations over them provide a uniform
framework for research works in numerous areas, and people have considered them for many special
cases, and obtained quite a few important results. However, it seems to us that a systematical
investigation of representations over diagrams of categories (specifically, abelian categories) is still
essential. In particular, some fundamental aspects of this topic need to be established:
(1) Describe the structure of D-Rep. For instances, under what conditions D-Rep is abelian or
Grothendieck ?
(2) Generalize important functors in representation theory and sheaf theory, and establish adjunc-
tion relations between them. For instances, by composing with a functor G:J→Iof small
categories, one can obtain a J-diagram of abelian categories from an I-diagram of abelian cat-
egories. Does this process always induce a functor between categories of representations, and
what are its left and right adjoint functors ? Similarly, does a morphism between two diagrams
of abelian categories always induce a functor between the categories of representations, and
what are its left and right adjoint functors ?
(3) Classify or characterize objects in the representation category with special homological prop-
erties such as projective objects, injective objects, flat objects, Gorenstein injective objects,
Gorenstein flat objects, etc.
(4) A “local-global” principle. That is, under what conditions can a certain special property (for
example, locally noetherian property) or a special structure (for example, a cotorsion pair or
an abelian model structure) shared by all local abelian categories Dibe amalgamated to the
corresponding property or structure of the global representation category D-Rep ?
The main goal of this paper is to study representations over diagrams of abelian categories
in a general framework, mostly focusing on the above mentioned aspects. As applications, for
some special index categories I, we construct corresponding global cotorsion pairs in D-Rep by
cotorsion pairs in each local abelian categories Di, investigate some important properties such as
hereditary and completeness of these global cotorsion pairs, and establish abelian model structures
on D-Rep. Furthermore, we classify or characterize objects with special homological properties
(such as projective, injective, flat, Gorenstein injective, and Gorenstein flat objects) in categories
of presheaves of modules over some combinatorial categories.
1which is a covariant functor from Ito the category of associative rings, and is called a representation of Iin that
paper. To avoid possible confusions, we call it an I-diagram of associative rings.
2We shall remind the reader that in our paper the abelian category Riis the left Ri-module category Ri-Mod
rather than the right Ri-module category Mod-Riused in [26].
4 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Structures of the category D-Rep.A fundamental aspect to understand D-Rep is to describe
its exact structure, that is, whether it is abelian or even Grothendieck. Under the assumptions that
the I-diagram Dof abelian categories is right exact (that is, Dαis right exact for any α∈Mor(I))
and strict (that is, Dis a functor from Ito the meta 2-category of abelian categories rather than a
pseudo-functor), Mozgovoy proved in [51] that D-Rep is abelian, and classified projective objects in
this category; see [51, Corollaries 2.9 and 2.13]. In [26], Estrada and Virili showed that under some
conditions the category R-Mod of left R-modules, where Ris an I-diagram of associative rings, is a
Grothendieck category, while it has a projective generator when Iis a poset; see [26, Theorem 3.18].
Our first main result shows that D-Rep is abelian or even Grothendieck under some relatively
weaker conditions, and hence lay a foundation for us to further consider homological properties
of representations over D; see Theorems 2.6 and 5.11. Moreover, it includes the above mentioned
results as special cases. In particular, we remove the conditions that Iis a poset and that Dis
strict, and hence our results can apply to a much more general situation.
Theorem A. Let Dbe a right exact I-diagram of abelian categories. Then D-Rep is an abelian
category. If further Dαpreserves small coproducts for any α∈Mor(I)and Diis a Grothendieck
category (resp., Grothendieck category with a family of projective generators) for any i∈Ob(I),
then D-Rep is a Grothendieck category (resp., Grothendieck category with a family of projective
generators).
Let Abe a cocomplete abelian category. Recall that an object Xin Ais said to be finitely
presented provided that the representable functor HomA(X, −) commutes with filtered colimits.
Denote the full subcategory of Aconsisting of finitely presented objects by Fp(A). The abelian
category Ais said to be locally finitely presented provided that Fp(A) is skeletally small and any
object in Ais a filtered colimit of finitely presented objects. Based on Theorem A, we prove that
if Dis a right exact I-diagram of abelian categories such that Dαpreserves small coproducts for
any α∈Mor(I) and Diis locally finitely presented for each i∈Ob(I), then D-Rep is locally
finitely presented; see Proposition 5.13. As a consequence of this fact, we obtain the following
representation theorem which is based on a general result of Crawley-Boevey [12, Theorem 1.4(2)].
Theorem B. Let Dbe a right exact I-diagram of abelian categories. Suppose that Dαpreserves
small coproducts for any α∈Mor(I)and Diis locally finitely presented for any i∈Ob(I). Then
D-Rep is equivalent to the subcategory of flat objects in the functor category Fun(Fp(D-Rep)op,Ab).
Functors and adjunctions. It is well known that a ring homomorphism ϕ:S→Tinduces a
functor from the category of T-modules to the category of S-modules, called the restriction functor
along ϕ. Moreover, it has a left adjoint called the induction functor and a right adjoint called
the coinduction functor. It is natural to ask whether this setup also holds for representations over
diagrams of abelian categories.
There are two distinct cases. For the first one, we consider a morphism F:D′→Dbetween
I-diagrams of abelian categories. Surprisingly, the induction functor (extension along F) always
exists, while we can only show the existence of the restriction functor under an extra condition. If
they both exist, they do form an adjoint pair; see Section 3. The second case is much more useful
for our purpose. That is, an I-diagram Dof abelian categories as well as a functor G:J→Iof
skeletally small categories induces a J-diagram D′of abelian categories, which is nothing but the
composite of Gand D. We show that under some mild conditions this process also induces another
restriction functor from D-Rep to D′-Rep, and it has a left adjoint. Explicitly, we establish the
following result; see Proposition 5.2 and Theorem 5.4.
Theorem C. Let Dbe a right exact I-diagram of abelian categories and Ja skeletally small
category. Then a functor G:J→Iinduces a J-diagram D′=D◦Gof abelian categories and an
exact restriction functor
G∗:D-Rep →D′-Rep.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 5
If further Disatisfies the axiom AB3 for any i∈Ob(I)and Dαpreserves small coproducts for any
α∈Mor(I), then Ginduces an extension functor
G!:D′-Rep →D-Rep,
which is a left adjoint of G∗.
Applying this theorem to the special case that Jis a full subcategory of Iconsisting of only one
object and Gis the natural inclusion functor, we get two functors called the evaluation functor and
the free functor. They play a crucial role in the proof of Theorem B as well as the classification of
projective objects in D-Rep.
Given a left ideal Iof a ring S, the restriction along ϕ:S→S/I is the familiar lift functor, via
which every S/I-module can be viewed as a S-module in a natural way. Analogously, we define
prime ideals Pof the morphism set of I: for any morphism αin Pand any morphism βin Mor(I),
its composite αβ or βα, whenever composable, is always contained in P; and furthermore, if a
composite αβ lies in P, then either αor βlies in Pas well. For each prime ideal P, we obtain a
subcategory I/Ptogether with an inclusion functor ιP:I/P→I; see Subsection 6.1.
Now we pause for a while to give an example illustrating the above construction. Let Xbe a
topological space and OXthe poset of open subsets in Xordered by set inclusions. Take a fixed
element x∈Xand define a subset Pxof Mor(OX) consisting of morphisms ι:U→Vin OX
such that either Uor Vdoes not contain x(clearly, this is equivalent to the condition that Udoes
not contain x). It is not hard to check that Pxis a prime ideal. Furthermore, OX/Pxis the full
subcategory of all open subsets containing x. This construction is used in sheaf theory to define
the stalk functor and its right adjoint, the skyscraper functor; see for instance [49, II.6].
Given a prime ideal Pand an I-diagram Dof abelian categories, as we mentioned before, we
obtain an I/P-diagram D◦ιPof abelian categories. Although I/Pis a subcategory rather than a
quotient category of I(compare to the above case of rings), we can still construct the lift functor
and its left adjoint, called the cokernel functor; see Propositions 6.1 and Theorem 6.3.
Theorem D. Let Pbe a prime ideal of the morphism set of Iand Da right exact I-diagram of
abelian categories. Then the inclusion functor ιPinduces a functor
lifP: (D◦ιP)-Rep →D-Rep.
If further Disatisfies the axiom AB3 for any i∈Ob(I/P)and Dαpreserves small coproducts for
any α∈Mor(I/P), then there exists a functor
cokP:D-Rep →(D◦ιP)-Rep
which is a left adjoint of lifP.
A special example of particular interest to us is as follows. Suppose that Iis a partially ordered
category3, that is, the relation i4jif and only if I(i, j ) is nonempty defines a partial order on
Ob(I). Fix i∈Ob(I). Then Pi=Mor(I)\EndI(i) is a prime ideal. We also suppose that Djsatisfies
both the axioms AB3 and AB3∗for every j∈Ob(I), and Dγpreserves small coproducts for any
endomorphism γin EndI(i). Then by the above theorem, we obtain two functors stai:Di→D-Rep
and coki:D-Rep →Disuch that (coki,stai) is an adjoint pair. We shall see that this adjoint pair
plays a key role for us to construct cotorsion pairs in D-Rep.
Cotorsion theory in D-Rep.The concept of cotorsion pairs was first introduced by Salce in [57]
and rediscovered by Enochs and Jenda in [21], which is analogous to the notion of torsion pairs
3In the literature some people call them directed categories or weakly directed categories. In this paper, since we
will consider direct categories defined in [42, Definition 5.1.1], to avoid possible confusion, we call them partially
ordered categories.
6 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
via replacing the Hom functor by the Ext functor. Let Abe an abelian category. A pair (C,D) of
subcategories of Ais called a cotorsion pair if C⊥=Dand ⊥D=C, where
C⊥={M∈A|Ext1
A(M, C ) = 0 for every object C∈C},and
⊥D={M∈A|Ext1
A(M, D) = 0 for every object D∈D}.
Following [21], a cotorsion pair (C,D) is said to be hereditary if it is both resolving and coresolving,
and complete if for any object Min A, there exist short exact sequences
0→D→C→M→0 and 0 →M→D′→C′→0
in Awith D, D′∈Dand C, C′∈C; see Section 8 for more details.
In the rest of this section, when considering D-Rep, we always impose the following assumptions:
Iis a partially ordered category, Dis a right exact I-diagram of Grothendieck categories
admitting enough projectives, and Dαpreserves small coproducts for all α∈Mor(I).
Fix i∈Ob(I) and consider the prime ideal Pi=Mor(I)\EndI(i) of Imentioned above. For any
left representation Xin D-Rep, by the universal property of colimits, one can construct a special
morphism ϕX
i: colimθ∈Pi(•,i)Dθ(Ms(θ))→Miin Di. Then for a family X={Xi}i∈Ob(I)with each
Xia subcategory of Di, we can define two subcategories of D-Rep as follows:4
Φ(X) = X∈D-Rep ϕX
iis a monomorphism and
coki(X) = coker(ϕX
i)∈Xifor all i∈Ob(I),and
D-RepX={X∈D-Rep |Xi∈Xifor all i∈Ob(I)}.
For details, please refer to Subsection 7.2. We mention that the definition of Φ(X) is inspired by
the classical works [8] by Birkhoff and [5] by Auslander and Reiten. Zhang refers to Φ(X) as a
monomorphism category; see [64].
Inspired by Holm and Jørgensen’s work [40, Theorem A], we establish a (hereditary) cotorsion
pair in D-Rep in Propositions 9.9 and 9.12 via gluing (hereditary) cotorsion pairs in those Di, when
the index category Iis a direct category defined in [42, Definition 5.1.1].
Theorem E. Suppose that Iis a direct category and Dis exact. Let X={Xi}i∈Ob(I)and Y=
{Yi}i∈Ob(I)be families of subcategories such that each (Xi,Yi)is a (hereditary) cotorsion pair in Di.
Then (Φ(X),D-RepY)is a (hereditary) cotorsion pair in D-Rep.
When Iis the free category associated to a left rooted quiver introduced in [23] (see Remark
7.6), we can obtain the following result: under a certain compatible condition the cotorsion pair
(Φ(X),D-RepY) established in Theorem E is complete if so is each cotorsion pair (Xi,Yi) in Di; see
Theorem 10.3. Here, a family X={Xi}i∈Ob(I)of subcategories is said to be compatible with respect
to Dif Dα(Xs(α))⊆Xt(α)for all α∈Mor(I) with source s(α) and target t(α).
Theorem F. Suppose that Iis the free category associated to a left rooted quiver and Dis exact.
Let X={Xi}i∈Ob(I)and Y={Yi}i∈Ob(I)be two families of subcategories such that (Xi,Yi)is a
complete cotorsion pair in Difor any i∈Ob(I)and Xis compatible with respect to D. Then
(Φ(X),D-RepY)is a complete cotorsion pair in D-Rep.
Remark. By Corollary 3.6, for every diagram Dof abelian categories indexed by a free category
I, one can construct a strict I-diagram D′of abelian categories such that the categories D-Rep and
D′-Rep are isomorphic, so one may suppose in the above theorem that Dis strict.
Remark. In [14, Theorem A], Di, Li, Liang and Xu first proved this result for categories of
representations of left rooted quivers, where they dropped the unnecessary condition imposed in
[53] that each cotorsion pair (Xi,Yi) in Diis hereditary.
4By abuse of notation, occasionally we use the same notation for both a subcategory and the class of objects of
this subcategory provided that it would not cause possible confusion. The reader can understand its precise meaning
from the context.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 7
Abelian model structures on D-Rep.Model category theory was introduced by Quillen in [55],
which provides a classical way of formally introducing homotopy theory into a category. Given a
bicomplete category Aand a class Wof morphisms in A, one may wish to treat the morphisms
in Was isomorphisms, that is, to understand the localization category A[W−1]. By imposing a
certain model structure on Aone can construct a new category Ho(A), the homotopy category of
A. A fundamental result shows that there is a canonical equivalence A[W−1]∼
=Ho(A); see [42].
A model category is said to be abelian if its underlying category Ais abelian and the model
structure is abelian, that is, it is compatible with the abelian structure of A. A celebrated result,
serving as a bridge connecting complete cotorsion pairs and abelian model structures, was estab-
lished by Hovey in [43], which is now known as Hovey’s correspondence: an abelian model structure
on Ais equivalent to a triple (Q,W,R) of subcategories of Asuch that Wis thick, and (Q,W∩R)
and (Q∩W,R) are two complete cotorsion pairs, where W(resp., Qand R) is the subcategory
of Aconsisting of all trivial (resp., cofibrant and fibrant) objects associated to the corresponding
abelian model structure. It is clear from Hovey’s correspondence that an abelian model structure
on Acan be succinctly represented by the triple (Q,W,R). Therefore, one often refers to such a
triple as an abelian model structure in the literature, and call it a Hovey triple. For more details
on abelian model structures, one refers to e.g. [7, 31, 32, 42, 43].
Recall that an abelian model structure (Q,W,R) on Ais said to be hereditary if both the
cotorsion pairs (Q,W∩R) and (Q∩W,R) are hereditary. Since most cotorsion pairs we encounter
are hereditary, Gillespie [30, Theorem 1.1] provided a more convenient way to construct abelian
model structures: if (Q,e
R) and (e
Q,R) are two hereditary and complete cotorsion pairs in Asuch
that e
Q⊆Q(or equivalently, e
R⊆R) and Q∩e
R=e
Q∩R, then there exists a unique thick subcategory
Wfor which (Q,W,R) is a Hovey triple, and this thick subcategory can be explicitly described.
As an application of Theorems E and F, we may construct a hereditary abelian model structure
on D-Rep from hereditary abelian model structures on those Di; see Theorem 10.9.
Theorem G. Suppose that Iis the free category associated to a left rooted quiver and Dis exact.
Then any family {(Qi,Wi,Ri)}i∈Ob(I)of hereditary abelian model structures on Disuch that both
Q={Qi}i∈Ob(I)and e
Q={Qi∩Wi}i∈Ob(I)are compatible with respect to Dinduces a hereditary
abelian model structure
(Φ(Q),D-RepW,D-RepR)
on D-Rep, where Wand Rdenote the families {Wi}i∈Ob(I)and {Ri}i∈Ob(I)of subcategories of Di
respectively.
Di, Estrada, Liang and Odaba¸sı established the above hereditary abelian model structures for
categories of representations of left rooted quivers [13, Theorem B]. The above theorem generalizes
their results to a much wider framework.
For a Grothendieck category Aadmitting enough projectives, a classical result by Hovey [43]
asserts that (Proj(A),A,A) is a hereditary abelian model structure on A, where Proj(A) denotes
the subcategory of Aconsisting of all projectives in A. Given an I-diagram Rof associative rings,
by Theorem A, we conclude that R-Rep is a Grothendieck category admitting enough projectives,
so one may ask whether Φ(Proj•) in Theorem G is actually the subcategory of projective objects
in R-Rep, where Proj•denotes the family {Proj(Ri-Mod)}i∈Ob(I)of subcategories. We answer this
question affirmatively in the following result and classify all projectives and flats in R-Rep for a
direct category I(e.g., the free category associated to a left rooted quiver); see Corollary 7.19 and
Theorem 11.5.
Theorem H. Suppose that Iis a direct category and Ris an I-diagram of associative rings. Then
Φ(Proj•)is actually the subcategory of projectives in R-Rep. Moreover, Φ(Flat•)is actually the
subcategory of flats in R-Rep, where Flat•denotes the family {Flat(Ri-Mod)}i∈Ob(I)of subcategories.
8 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Gorenstein flat model structure on R-Rep.For finitely generated modules over a Noetherian
ring, Auslander and Bridger [3] introduced a homological invariant called the G-dimension, which
was soon generalized to arbitrary modules. Enochs, Jenda and Torrecillas proposed in [20] and [22]
two primary generalized classes of modules of G-dimension 0: Gorenstein projective modules and
Gorenstein flat modules. They were found to be of particular importance and were further treated
by Holm in [39]. For an arbitrary ring A,ˇ
Saroch and ˇ
St’ov´ıˇcek introduced a new subcategory
of projectively coresolved Gorenstein flat A-modules in [62], which allows them to construct a
new abelian model structure. Specifically, let GF(A) (resp., PGF(A)) denote the subcategory of
Gorenstein flat (resp., projectively coresolved Gorenstein flat) A-modules. ˇ
Saroch and ˇ
St’ov´ıˇcek
proved that there exists a hereditary abelian model structure (GF(A),PGF(A)⊥,Cot(A)), where
Cot(A) is the subcategory of cotorsion A-modules. In particular, if Ris an I-diagram of associative
rings, their construction yields a hereditary abelian model structure
(GF(Ri-Mod),PGF(Ri-Mod)⊥,Cot(Ri-Mod))
on Ri-Mod for every i∈Ob(I). On the other hand, it follows from Lemma 11.22 that both
the families Flat•={Flat(Ri-Mod)}i∈Ob(I)and GF•={GF(Ri-Mod)}i∈Ob(I)of subcategories are
compatible with respect to the associated I-diagram Rof module categories. Thus as an application
of Theorem G, we get a hereditary model structure on the category R-Rep and classify all Gorenstein
flat objects and cotorsion objects in R-Rep; see Theorems 11.31 and 11.28, Proposition 11.25 and
Corollary 11.30. Here, the symbols PGF•and Cot•denote the families {PGF(Ri-Mod)}i∈Ob(I)and
{Cot(Ri-Mod)}i∈Ob(I)of subcategories, respectively.
Theorem I. Suppose that Iis the free category associated to a left rooted quiver and Ris a flat
I-diagram of associative rings such that
(a) Riis commutative for any i∈Ob(I), or
(b) Rjhas finite projective dimension as a left Ri-module for any α:i→jin Mor(I).
Then there exists a hereditary abelian model structure
(Φ(GF•),R-RepPGF⊥
•,R-RepCot•)
on R-Rep. Moreover, Φ(GF•) (resp., R-RepCot•)is the subcategory of Gorenstein flat (resp.,
cotorsion)objects in R-Rep, and R-RepPGF⊥
•is the right orthogonal subcategory PGF(R-Rep)⊥.
Gorenstein injective model structure on Rep-R.The category Rep-Dof right representations
over the I-diagram Dof abelian categories is defined to have objects Nconsisting of the following
data:
•for any i∈Ob(I), an object Niin Di, and
•for any α:i→jin Mor(I), a morphism Nα:Nj→Dα(Ni) in Dj
such that two axioms are satisfied; see Definition 2.13 for details. Dual to Theorem 2.6, we see that
if Dis left exact, then Rep-Dis an abelian category. If further Iis a partially ordered category and
Dαpreserves small products for any α∈Mor(I), then for any i∈Ob(I), there exists a kernel functor
keri:Rep-D→Di(see Subsection 6.3), which is dual to the cokernel functor coki:D-Rep →Di.
Consider again the prime ideal Pi=Mor(I)\EndI(i) of Imentioned before. For any right
representation Yin Rep-D, by the universal property of limits, one can construct a special morphism
ψY
i:Yi→limθ∈Pi(•,i)Dθ(Ys(θ)) in Di. Then for a family Y={Yi}i∈Ob(I)with each Yia subcategory
of Di, we can define two subcategories of Rep-D:
Ψ(Y) = Y∈Rep-DψY
iis an epimorphism and
keri(Y) = ker(ψY
i)∈Yifor all i∈Ob(I),and
(Rep-D)Y={Y∈Rep-D|Yi∈Yifor all i∈Ob(I)}.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 9
By this construction, one can obtain dual versions of Theorems E, F and G for the subcategory
Ψ(Y).
According to Subsection 4.1, an I-diagram Rof associative rings induces a left exact I-diagram
Rof module categories with
•Ri=Mod-Ri, the category of right Ri-modules for any i∈Ob(I), and
•Rα= HomRi(Rj,−) for any α:i→jin Mor(I).
In particular, dual to Proposition 4.5, we concludes that Rep-Rcoincides with the category
Mod-Rof right R-modules in the sense of Estrada and Virili [26]. Denote by Inj•the family
{Inj(Mod-Ri)}i∈Ob(I)of subcategories. Then dual to Theorem H, we have the following result class-
fying all injectives in Rep-R; see Corollary 7.23.
Theorem J. Suppose that Iis a direct category and Ris an I-diagram of associative rings. Then
Ψ(Inj•)is the subcategory of injectives in Rep-R.
Finally, we get the following hereditary abelian model structure on Rep-Rand classify all Goren-
stein injective objects in Rep-R; see Theorem 11.16 and Corollaries 11.18 and 11.19. Here, denote
by GI•the family {GI(Mod-Ri)}i∈Ob(I)with each GI(Mod-Ri) the subcategory of Mod-Riconsisting
of Gorenstein injective objects.
Theorem K. Suppose that Iis the free category associated to a left rooted quiver and Ris a flat
I-diagram of associative rings satisfying condition (a)or (b)in Theorem I. Then there exists a
hereditary abelian model structure
(Rep-R,(Rep-R)⊥GI•,Ψ(GI•))
on Rep-R. Moreover, Ψ(GI•)is the subcategory of Gorenstein injective objects in Rep-R.
Remark. At a first glance, Theorem K is dual to Theorem I, but it is actually not. It is surprising
to us that these two theorems share the same assumptions, that is, we assume in both theorems
that Ris a flat I-diagram of associative rings satisfying
(a) Riis commutative for any i∈Ob(I), or
(b) Rjhas finite projective dimension as a left Ri-module for any α:i→jin Mor(I).
Actually, condition (a) or (b) guarantees that the family GI•is compatible with respect to R(see
Lemma 11.10), which is crucial to study both Gorenstein injective objects in Rep-Rand Gorenstein
flat objects in R-Rep; while the condition that Ris a flat I-diagram of associative rings ensures
that the I-diagram Rof module categories is exact in Theorem I, as Rα:Ri-Mod →Rj-Mod is
nothing but the tensor product functor Rj⊗Ri−for any α:i→jin Mor(I). The reader may
expect to replace the condition that Ris flat in Theorem I by the condition that Ris projective to
make sure that the I-diagram Rof module categories in Theorem K is exact, as in the second case
Rα= HomRi(Rj,−) : Mod-Ri→Mod-Rjfor any α:i→jin Mor(I). However, it turns out that
we do not have to make this change in Theorem K.
We end the introduction with the following notations and conventions.
Notations and Conventions. Throughout the paper,
•let Ibe a skeletal small category with the set of objects Ob(I) and the set of morphisms
Mor(I);
•for α∈Mor(I), write s(α) for its source and t(α) for its target;
•for i∈Ob(I), denote by eithe identity morphism on i, and denote by EndI(i) the set of
endomorphisms;
•let Iibe the subcategory of Iconsisting of the single object iand the single morphism ei;
•by the term “subcategory” in an abelian category, we always mean a full subcategory which
is closed under isomorphisms and contains the zero object;
10 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
•a functor between categories (resp., abelian categories) is always assumed to be covariant
(resp., additive and covariant);
•denote by AB the meta 2-category5of abelian categories, while by Ab the full subcategory
of AB consisting of abelian groups;
•denote by Ring (resp., Comm.Ring) the category of associative (resp., commutative) rings;
•given an abelian category A, denote by Proj(A) (resp., Flat(A), Cot(A), GF(A), PGF(A),
Inj(A), GI(A)) the subcategory of Aconsisting of projective (resp., flat, cotorsion, Gorenstein
flat, projectively coresolved Gorenstein flat, injective, Gorenstein injective) objects;
•denote by the symbol (−)+the contravariant functor HomZ(−,Q/Z).
5By a meta 2-category we mean that the functors between two arbitrary objects in this “category” form a class,
not necessarily a set.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 11
Part I. Diagrams and their representations
The first part of this paper serves as an introduction to diagrams of categories and representations
over them. We introduce definitions, examples, and elementary results, which will be extensively
used in the rest of this paper. As an application as well as an important example, we reformulate
modules over presheaves of associative rings studied by Estrada and Virili [26] in terms of our
notions.
1. Diagrams of categories
In this section, we introduce I-diagrams of categories and morphisms between them, and construct
some special morphisms which will be used to explore relations between categories of representations
over I-diagrams in the next section.
We begin with the notion of Godement products of functors. Let F1, F2:S→Tand G1, G2:
T→Ube four functors between categories, and let α:F1→F2and β:G1→G2be two
natural transformations. Recall that the Godement product (see e.g. [9]) of αand βis the natural
transformation
β∗α:G1◦F1→G2◦F2
defined by setting (β∗α)(S) to be
(1.0.1) β(F2(S)) ◦G1(α(S)) = G2(α(S)) ◦β(F1(S))
for any object Sin S. It is routine to check that the operation “∗” is associative.
Now we give the definition of I-diagrams of categories, a central concept studied in this paper.
1.1 Definition. An I-diagram of categories is a tuple (D, η, τ ) (frequently denoted by Dfor brevity
when ηand τare clear from the context) consisting of the following data:
•for every i∈Ob(I), a category Di,
•for every α:i→jin Mor(I), a functor Dα:Di→Dj,
•for every i∈Ob(I), a natural isomorphism ηi: idDi
≃
−→ Dei, and
•for any pair of composable morphisms αand βin Mor(I), a natural isomorphism
τβ,α :Dβ◦Dα≃
−→ Dβα
such that the following two axioms are satisfied:
(Dia.1) Given composable morphisms iα
→jβ
→kγ
→lin Mor(I), there exists a commutative diagram
Dγ◦Dβ◦Dα
idDγ∗τβ,α
//
τγ,β ∗idDα
Dγ◦Dβα
τγ,βα
Dγβ ◦Dα
τγβ,α
//Dγβα
of natural isomorphisms.
(Dia.2) Given a morphism iα
→jin Mor(I), there exists a commutative diagram
Dα
idDα∗ηi
zz✈
✈
✈
✈✈
✈
✈
✈✈ηj∗idDα
$$
■
■
■
■
■
■
■
■
■
Dα◦Dei
τα,ei$$
❍
❍
❍
❍
❍
❍
❍
❍
❍
Dej◦Dα
τej,α
zz✉
✉
✉
✉
✉
✉
✉
✉
✉
Dα
of natural isomorphisms.
12 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
An I-diagram (D, η, τ ) of categories is said to be strict if ηiis the identity for any i∈Ob(I), and
τβ,α is the identity for any pair of composable morphisms αand βin Mor(I).
1.2 Remark. The categorically inclined reader may realize that an I-diagram of categories is
exactly a pseudo lax functor from I, viewed naturally as a 2-category, to the meta 2-category CAT
consisting of all categories (see Street [59]), while a strict I-diagram is exactly a 2-functor from Ito
CAT. Note that an I-diagram of categories is also called an I-indexed category by Johnstone [45].
We call (D, η, τ ) an I-diagram of abelian categories if Diis an abelian category for each i∈Ob(I)
and the functor Dαis additive and covariant for any α∈Mor(I).
1.3 Definition. Let Dbe an I-diagram of abelian categories. Then Dis said to be
(a) exact (resp., left exact, right exact) if the functor Dαis exact (resp., left exact, right exact)
for any α∈Mor(I);
(b) admitting enough right adjoints (resp., left adjoints ) if the functor Dαadmits a right adjoint
(resp., left adjoint) for any α∈Mor(I);
(c) endo-trivial if for any i∈Ob(I) and any endomorphism γ∈EndI(i), there exists a natural
isomorphism ηγ
i: idDi→Dγ;
(d) locally exact if for any i∈Ob(I) and any endomorphism γ∈EndI(i), the functor Dγis exact.
Clearly, if Dis endo-trivial, then it is locally exact.
1.4 Remark. Give an abelian category A, if we let Di=Afor all i∈Ob(I) and Dαbe the identity
functor on Afor all α∈Mor(I), then Dis called a trivial I-diagram of A. It is clear that trivial
I-diagrams satisfy all conditions in the above definitions, and coincide with covaraint functors from
Ito A. We will give in Sections 2 and 4 a few non-trivial examples of diagrams of abelian categories
satisfying certain conditions in the above definition. For instances, the diagram Uin Example 2.4
is strict, exact and admits both enough right and left adjoints. Both the diagrams Rand Rin
Subsection 4.1 are endo-trivial. Meanwhile, Ris right exact and admits enough right adjoints, and
Ris left exact and admits enough left adjoints.
The following result shows that any I-diagram of categories admitting enough adjoints induces
an Iop-diagram of categories, where Iop is the opposite category of I.
1.5 Proposition. Let Dbe an I-diagram of categories. If Dadmits enough right adjoints (resp.,
left adjoints), then it induces an Iop-diagram L(resp., K) admitting enough left adjoints (resp.,
right adjoints).
Proof. We only prove the statement for the case that Dadmits enough right adjoints since the
other case follows by a dual argument. The definition of Lis straightforward as follows:
•for i∈Ob(Iop), set Lito be Di;
•for αop :j→iin Mor(Iop), set Lαop to be the right adjoint of Dα;
•for i∈Ob(Iop), since both (idDi,idDi) and (Dei,Lei) are adjoint pairs and there is a natural
isomorphism ηi: idDi→Dei, one can obtain a natural isomorphism µi: idLi= idDi→Lei;
•for any pair of composable morphisms βop :k→jand αop :j→iin Mor(Iop), note
that both (Dβ◦Dα,Lαop ◦Lβop ) and (Dβα,Lαopβop ) are adjoint pairs and there exists
a natural isomorphism τβ,α :Dβ◦Dα→Dβα, one can obtain a natural isomorphism
ναop,βop :Lαop ◦Lβop →Lαopβop .
According to the above setting, it is routine to check that the triple (L, µ, ν ) satisfies the axioms
(Dia.1) and (Dia.2) in Definition 1.1 by Yoneda’s Lemma since so does (D, η, τ ). Moreover, note
that Lαop has a left adjoint Dαfor any αop ∈Mor(Iop). Thus, Lis an Iop-diagram of categories
admitting enough left adjoints.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 13
In order to compare two I-diagrams of categories and consider further the relationship between
representations over them, we introduce morphisms between two I-diagrams of categories.
1.6 Definition. Let (D′, η′, τ ′) and (D, η, τ ) be two I-diagrams of categories. A morphism Ffrom
D′to Dconsists of the following data:
•for any i∈Ob(I), a functor Fi:D′
i→Di, and
•for any α:i→jin Mor(I), a natural isomorphism Fα:Fj◦D′
α→Dα◦Fi
such that the following two axioms are satisfied:
(Mor.1) Given composable morphisms iα
→jβ
→kin Mor(I), there exists a commutative diagram
Fk◦D′
β◦D′
α
Fβ∗idD′
α//
idFk∗τ′
β,α
Dβ◦Fj◦D′
α
idDβ∗Fα
//Dβ◦Dα◦Fi
τβ,α ∗idFi
Fk◦D′
βα
Fβα
//Dβα ◦Fi
of natural isomorphisms.
(Mor.2) Given an object i∈Ob(I), there exists a commutative diagram
Fi
ηi∗idFi//
idFi∗η′
i##
❋
❋
❋
❋
❋
❋
❋
❋
❋Dei◦Fi
Fi◦D′
ei
Fei
99
s
s
s
s
s
s
s
s
s
s
of natural isomorphisms.
Setup. Throughout this paper, when we consider a morphism Fbetween I-diagrams of abelian
categories, we always suppose that the functor Fiis additive and covariant for i∈Ob(I).
1.7 Remark. The categorically inclined reader may realize that a morphism between two I-
diagrams of categories is exactly a pseudo natural transformation (also called an I-indexed functor
by Johnstone in [45]).
Under a certain condition, a morphism between I-diagrams admitting enough adjoints yields a
morphism between the induced Iop-diagrams.
1.8 Proposition. Let F:D′→Dbe a morphism of I-diagrams of categories such that both D′
and Dadmit enough right adjoints. If Fiadmits a right adjoint for any i∈Ob(I), then Finduces
a morphism G:L→L′between the induced Iop-diagrams given in Proposition 1.5.
Proof. It follows from Proposition 1.5 that the I-diagram D′(resp., D) induces an Iop-diagram
L′(resp., L), which admits enough left adjoints. Thus the definition of Gis straightforward as
follows:
•for i∈Ob(Iop), set Gi:Li→L′
ito be the right adjoint of Fi.
•for αop :j→iin Mor(Iop), note that Gi◦Lαop and L′
αop ◦Gjare the right adjoint of
Dα◦Fiand Fj◦D′
α, respectively. Since Fα:Fj◦D′
α→Dα◦Fiis a natural isomorphism
(see Definition 1.6), there exists a natural isomorphism
Gαop :Gi◦Lαop →L′
αop ◦Gj.
It is routine to check that Gsatisfies the axioms (Mor.1) and (Mor.2) in Definition 1.6 by
Yoneda’s Lemma since so does F. Thus G:L→L′is a morphism between Iop-diagrams.
Dual to Proposition 1.8, we have:
14 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
1.9 Proposition. Let F:D′→Dbe a morphism between I-diagrams of categories such that
both D′and Dhave enough left adjoints. If Fiadmits a left adjoint for any i∈Ob(I), then F
induces a morphism E:K→K′of the induced Iop-diagrams given in Proposition 1.5.
In the special case that Iis the free category associated to a quiver, we show that any I-diagram
of (abelian) categories induces a corresponding strict I-diagram of (abelian) categories, and there
exists a special morphism between these two I-diagrams. Furthermore, we will show in the next
section that the categories of representations over these two I-diagrams are isomorphic. We begin
with a lemma, which is from [9, Proposition 1.3.5].
1.10 Lemma. Let F1, F2, F3:S→Tand G1, G2, G3:T→Ube six functors between categories,
and let α:F1→F2,γ:F2→F3,β:G1→G2and δ:G2→G3be four natural transformations.
Then one has
(δ∗γ)◦(β∗α) = (δ◦β)∗(γ◦α).
1.11 Proposition. Let Ibe the free category associated to a quiver and (D, η, τ )an I-diagram of
(abelian) categories. Then there exist a strict I-diagram D′of (abelian) categories and a morphism
F:D′→Dsuch that Fiis the identity for every i∈Ob(I).
Proof. Suppose that Iis associated to the quiver Q= (Q0, Q1), where Q0is the set of vertices
and Q1is the set of arrows. Define D′as follows:
•for i∈Ob(I), set D′
ito be Di;
•for the identity eion i∈Ob(I), set D′
eito be idDi;
•for a non-identity morphism α:i→jin Mor(I), which can be expressed uniquely as
αl◦αl−1◦ · · · ◦ α1with α1,··· , αlarrows in Q1(l=l(α) is called the length of α), set the
functor D′
α:Di→Djto be the composite Dαl◦Dαl−1◦ · · · ◦ Dα1.
It is evident that D′is a strict I-diagram of (abelian) categories.
Next, we define an assignment F:D′→Das follows:
•for i∈Ob(I), set Fito be idDi;
•for α∈Mor(I), set6
Fα=
ηi,if α=ei,
idDα,if l= 1,
ταl,αl−1◦···◦α1◦(idDαl∗ταl−1,αl−2◦···◦α1)◦ · · · ◦ (idDαl◦···◦Dα3∗τα2,α1) if l>2.
It is clear that Fαis a natural isomorphism for any α∈Mor(I). To show that Fis a morphism
between I-diagrams of abelian categories, we only need to prove that {Fα}α∈Mor(I)satisfies the
axioms (Mor.1) and (Mor.2) in Definition 1.6.
The axiom (Mor.2) clearly holds since Fi= idDi,Fei=ηiand D′is a strict I-diagram of (abelian)
categories. It remains to show that {Fα}α∈Mor(I)satisfies the axiom (Mor.1), that is,
τβ,α ◦(Fβ∗Fα) = Fβα (‡)
for any pair of composable morphisms iα
→jβ
→kin Mor(I). We check it case by case.
(a) If α=ei, then
τβ,ei◦(Fβ∗Fei) = τβ,ei◦(Fβ∗ηi) = (idDβ∗η−1
i)◦(Fβ∗ηi) = (idDβ◦Fβ)∗(η−1
i◦ηi) = Fβ,
where the third equality holds by Lemma 1.10. A similar argument also works for the case
that β=ej.
6If l= 2 then Fα=τα2,α1; if l= 3 then Fα=τα3,α2◦α1◦(idDα3∗τα2,α1).
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 15
(b) If l(α) = 1 = l(β), then
τβ,α ◦(Fβ∗Fα) = τβ,α ◦(idDβ∗idDα) = τβ,α ◦idDβ◦Dα=τβ,α =Fβα ,
where the first and fourth equalities hold by relevant definitions.
(c) If l(α) = n>2 and l(β) = 1, a detailed check of (‡) is given in Appendix B (see B.1).
(d) If l(α) = n>2 and l(β) = m>1, then we proceed by an induction on m. The case m= 1 has
been handled in (c). Suppose now that (‡) holds for m−1 and let β=βm◦β′with l(βm) = 1
and l(β′) = m−1. A detailed check of (‡) is given in B.1.
This completes the proof.
2. Representations over diagrams of categories
In this section we introduce the definition of left (right) representations over an I-diagram of
categories, and give the abelian structure for categories of representations.
2.1. Left representations over diagrams of categories. We begin with the following definition.
2.1 Definition. Let (D, η, τ ) be an I-diagram of categories. A left representation Mover D
consists of the following data:
•for every i∈Ob(I), an object Miin Di, and
•for every α:i→jin Mor(I), a structural morphism Mα:Dα(Mi)→Mjin Dj
such that the following two axioms are satisfied:
(lRep.1) Given composable morphisms iα
→jβ
→k∈Mor(I), there exists a commutative diagram
Dβα(Mi)Mβα
//Mk
Dβ(Dα(Mi))
τβ,α(Mi)
OO
Dβ(Mα)
//Dβ(Mj)
Mβ
OO
in Dk, that is, Mβα ◦τβ,α(Mi) = Mβ◦Dβ(Mα).
(lRep.2) Given an object i∈Ob(I), there exists a commutative diagram
Mi
idMi//
ηi(Mi)$$
❍
❍
❍
❍
❍
❍
❍
❍
❍Mi
Dei(Mi)
Mei
::
✈
✈
✈
✈
✈
✈
✈
✈
✈
in Di, that is, Mei=η−1
i(Mi).
A morphism ω:M→M′between two left representations Mand M′over Dis a family
{ωi:Mi→M′
i}i∈Ob(I)of morphisms such that the diagram
Dα(Mi)Dα(ωi)
//
Mα
Dα(M′
i)
M′
α
Mj
ωj
//M′
j
in Djcommutes for any α:i→j∈Mor(I).
Denote by D-Rep the category of all left representations over D. In the situation that Dis an
I-diagram of abelian categories, it is clear that D-Rep is an additive category with the zero object.
From now on, we always assume that Dis an I-diagram of abelian categories. We give a few
examples of left representations over D.
16 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
2.2 Example. Let F:C→Dbe a right exact functor between abelian categories Cand D.
One can construct an abelian category, denoted by (F↓D), whose objects are the morphisms
σ:F(C)→Dwith C∈Cand D∈D, and morphisms from the object σ:F(C)→Dto the object
σ′:F(C′)→D′are the pair (α:C→C′, β :D→D′) of morphisms such that β◦σ=σ′◦F(α).
Such a category is called a comma category in the literature. Examples of comma categories include
module categories over triangular matrix rings, morphism categories of abelian categories, etc.
Let Ibe the free category associated to the quiver (1 α
→2). One can define a strict right exact
I-diagram Dof abelian categories with D1=C,D2=Dand Dα=F. It is evident that the comma
category (F↓D) coincides with the category D-Rep.
2.3 Example. Recall that a Morita context ring is a matrix
Λ = R M
N S
where Rand Sare two associative rings, Mis an (S, R)-bimodule, and Nis an (R, S )-bimodule,
together with a morphism
φ:N⊗SM→R
of (R, R)-bimodules and a morphism
ψ:M⊗RN→S
of (S, S)-bimodules. A module over Λ is a quadruple (X, Y, f, g) where Xis an R-module, Yis an
S-module,
f:M⊗RX→Y
is an S-module homomorphism, and
g:N⊗SY→X
is an R-module homomorphism such that certain compatibility conditions are satisfied. For details,
please refer to [35].
We observe that the module category of Λ can be realized as D-Rep of a special diagram Dof
abelian categories. Explicitly, let Ibe the free category associated to the following quiver:
x
α
((y
β
hh
and let Dx=R-Mod,Dy=S-Mod,Dα=M⊗R−, and Dβ=N⊗S−. The reader can check that
a left representation over Dis precisely a Λ-module, so Λ-Mod coincides with D-Rep.
2.4 Example. Gao, K¨ulshammer, Kvamme and Psaroudakis introduced in [28] the notion of a
phylum on a quiver Q= (Q0, Q1) as an extension of Gabriel’s notion of species [27]. Recall from
[28, Definition 4.1] that a phylum Uon Qconsists of the following data:
•for a vertex i∈Q0, an abelian category Ai;
•for an arrow α:i→jin Q1, a pair of functors (Fα:Ai→Aj, Gα:Aj→Ai) such that
both (Fα, Gα) and (Gα, Fα) are adjoint pairs.
Examples of representations over phyla include modules over triangular matrix rings, etc; see [28,
Example 4.5] and [46, Example 2.10(c)] for details.
Given a phylum Uon Q, define Uas follows:
•for i∈Q0, set Uito be Ai;
•for the identity eion i, set Ueito be idAi;
•for any non-trivial path γ:i→j, since γcan be written uniquely as γl◦γl−1◦ · · · ◦ γ1with
all γ1,··· , γlarrows in Q1, set the functor Uγ:Ai→Ajto be Fγl◦Fγl−1◦ · · · ◦ Fγ1.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 17
It is evident that Uis a strict exact I-diagram of abelian categories, where Iis the free category
associated to Q. Furthermore, the category Rep(U) of U-representations given in [28, Definition 4.4]
coincides with the category U-Rep.
2.5 Example. Estrada and Virili introduced in [26] representations of small categories, which are
diagrams of associative rings in our sense and can be viewed as a common generalization of some
important algebraic structures naturally arising in geometric contexts. In Section 4, we will show
that the category of left R-modules in the sense of Estrada and Virili, where Ris a diagram of
associative rings, coincides with the category R-Rep for a certain I-diagram Rof module categories
induced by R.
The cateogry D-Rep is in general not abelian. However, under a mild assumption, we can show:
2.6 Theorem. Let (D, η, τ )be a right exact I-diagram of abelian categories. Then D-Rep is an
abelian category. Moreover, a sequence M→N→Kin D-Rep is exact if and only if Mi→Ni→
Kiis exact in Difor each i∈Ob(I).
Proof. Let ω:M→M′be a morphism in D-Rep. We construct the cokernel Cok(ω) of ωas
follows. For i∈Ob(I), let πi:M′
i→coker(ωi) be the cokernel of ωi:Mi→M′
i. For α:i→jin
Mor(I), consider the commutative diagram
(2.6.1) Dα(Mi)Dα(ωi)
//
Mα
Dα(M′
i)
M′
α
Dα(πi)
//Dα(coker(ωi)) //
coker(ω)α
0
Mj
ωj
//M′
j
πj
//coker(ωj)//0
in Dj. Since Dαis right exact by assumption, the first row in the above commutative diagram is
exact. Consequently, Dα(πi) is the cokernel of Dα(ωi). Note that
πj◦M′
α◦Dα(ωi) = πj◦ωj◦Mα= 0.
By the universal property of the cokernel, there exists a morphism coker(ω)αsuch that the right
square in the above diagram commutes. Thus, we can make the following construction:
•for i∈Ob(I), set Cok(ω)i= coker(ωi), and
•for α:i→jin Mor(I), set Cok(ω)α= coker(ω)α.
It is routine to check that Cok(ω) satisfies the axioms (lRep.1) and (lRep.2), and hence, is an object
in D-Rep. For details, please refer to Subsection B.2.
We have already obtained a morphism π:M′→Cok(ω) in D-Rep such that π◦ω= 0. Let us
show that πis the cokernel of ω. Choose a morphism ǫ:M′→M′′ in D-Rep such that ǫ◦ω= 0.
For any i∈Ob(I), note that πiis the cokernel of ωiand ǫi◦ωi= 0. Consequently, there is a unique
morphism ρi: Cok(ω)i→M′′
isuch that the diagram
(2.6.2) Mi
ωi//M′
i
ǫi
##
●
●
●
●
●
●
●
●
●
πi//Cok(ω)i
ρi
M′′
i
18 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
commutes. To see that ρ={ρi}i∈Ob(I)is a morphism in D-Rep, we need to verify that the right
square in the diagram
Dα(M′
i)Dα(πi)
//
M′
α
Dα(Cok(ω)i)
Cok(ω)α
Dα(ρi)
//Dα(M′′
i)
M′′
α
M′
j
πj
//Cok(ω)j
ρj
//M′′
j
commutes for any α:i→jin Mor(I); that is,
M′′
α◦Dα(ρi) = ρj◦Cok(ω)α.(‡)
This is checked in Subsection B.2.
We can construct the kernel of ωdually, with the only difference that no exactness conditions
of Dis required. Moreover, it is also clear that the canonical morphism between the image and
the coimage is an isomorphism as it is an isomorphism on each component. Consequently, D-Rep
is an abelian category. The last statement is obvious by the definition of morphisms in D-Rep; see
Definition 2.1.
2.2. Limits and colimits. In this subsection we study limits and colimits in D-Rep, which have
been considered in [44, Subsection 2.3] for the special circumstance that Dadmits enough right
adjoints. The reader can see that constructions and properties described in this subsection are
quite similar to the ones in that paper. Firstly, we recall the AB axioms.
2.7 Definition. An abelian category Asatisfies the axiom AB3 if it is cocomplete, that is, it has
small coproducts. The abelian category Asatisfies the axiom AB4 if it satisfies the axiom AB3
and the further condition that any small coproduct of monomorphisms is a monomorphism. The
axioms AB3∗and AB4∗are dual to the axioms AB3 and AB4, respectively. Note that the axiom
AB3 is equivalent to the existence of arbitrary small colimits. We say that Asatisfies the axiom
AB5 if it satisfies the axiom AB3 and the condition that filtered colimits are exact.
A set Gof objects in Ais called a family of generators provided that for any non-zero morphism
f:X→Yin A, there exist an object G∈Gand a morphism g:G→Xsuch that f g 6= 0. When
Asatisfies the axiom AB5 and possesses a family of generators, it is called a Grothendieck category.
2.8 Remark. Note that any colimit can be realized as the cokernel of a morphism of two small
coproducts. For a functor F:A′→Abetween two abelian categories which both have small
coproducts, if Fis right exact and preserves small coproducts, then it preserves small colimits
as well. Dually, if both A′and Ahave small products, and Fis left exact and preserves small
products, then it preserves small limits.
Let Dbe a right exact I-diagram of abelian categories such that Diadmits small coproducts for
every i∈Ob(I) and Dαpreserves small coproducts for every α∈Mor(I), that is, the morphism κα:
`u∈UDα(Mu
i)→Dα(`u∈UMu
i) induced by the universal property of coproducts is an isomorphism
for any indexed set U. For a family {Mu}u∈Uof objects in D-Rep, the coproduct `u∈UMuof
{Mu}u∈Ucan be taken componentwise. Explicitly,
•for i∈Ob(I), (`u∈UMu)i=`u∈UMu
i;
•for α:i→jin Mor(I), by the universal property of the coproducts, there exists a unique
morphism ςαsuch that the diagram
Dα(Mu
i)Mu
α//
ε
Mu
j
ε′
`u∈UDα(Mu
i)ςα//`u∈UMu
j= (`u∈UMu)j
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 19
commutes, where εand ε′are the canonical injections. Since Dαpreserves small coproducts
by assumption, the structural morphism (`u∈UMu)αis defined to be the composite
ςα◦κ−1
α:Dα(( a
u∈U
Mu)i)→(a
u∈U
Mu)j.
The reader can check that the above setting of `u∈UMusatisfies axioms (lRep.1) and (lRep.2),
and hence, is an object in D-Rep. Furthermore, it satisfies the universal property of coproducts.
2.9 Remark. Note that if Dsatisfies the conditions specified in the construction of coproducts,
then Dihas small colimits for any i∈Ob(I) and Dαpreserves small colimits for any α∈Mor(I);
see Remark 2.8. Therefore, given a direct system ((Mx),(fyx)) of objects in D-Rep, by a similar
method one can construct the colimit componentwise.
The following result gives sufficient conditions such that the category D-Rep satisfies AB-axioms.
2.10 Proposition. Let Dbe a right exact I-diagram of abelian categories such that Dαpreserves
small coproducts for any α∈Mor(I). If Disatisfies the axiom AB3 (resp., AB4,AB5)for any
i∈Ob(I), then so does D-Rep.
Proof. By Theorem 2.6, we see that D-Rep is an abelian category. Note that the exactness of
a sequence in D-Rep is completely determined by the exactness of the corresponding sequences
obtained by restricting it to objects in Difor i∈Ob(I) (see Theorem 2.6). By the descriptions of
coproducts and colimits in D-Rep, it follows that if Disatisfies the axiom AB3 (resp., AB4 or AB5)
for any i∈Ob(I), so does D-Rep.
Now we describe products and limits in D-Rep. Let Dbe a right exact I-diagram of abelian
categories such that Dihas small products for each i∈Ob(I). For a family {Mu}u∈Uof objects
in D-Rep with Uthe index set, analogue to the coproduct `u∈UMu, one can define the product
Qu∈UMuof {Mu}u∈Ucomponentwise. Explicitly,
•for i∈Ob(I), (Qu∈UMu)i=Qu∈UMu
i;
•for α:i→jin Mor(I), by the universal property of products, there exists a unique
morphism σαsuch that the diagram
Qu∈UDα(Mu
i)σα//
π
Qu∈UMu
j= (Qu∈UMu)j
π′
Dα(Mu
i)Mu
α//Mu
j
commutes, where πand π′are the canonical projections. The structural morphism (Qu∈UMu)α
is defined to be the composite
σα◦ια:Dα(( Y
u∈U
Mu)i)→(Y
u∈U
Mu)j,
where ια:Dα(Qu∈UMu
i)→Qu∈UDα(Mu
i) is the morphism induced by the universal
property of products.
The reader can check that Qu∈UMuis indeed an object in D-Rep, and satisfies the universal
property of products.
2.11 Remark. Note that if Dsatisfies the conditions specified in the construction of products,
then Dihas small limits for any i∈Ob(I) since any limit can be realized as the kernel of a morphism
of two small products. Therefore, given an inverse system ((Mx),(fxy)) of objects in D-Rep, by
a similar method one can define the limit componentwise. We also point out a small difference
between the constructions of coproducts and products. Explicitly, when defining coproducts and
20 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
colimits, we have to assume that Dαpreserves small coproducts for every α∈Mor(I). However,
when defining products and limits, we don’t have to assume that Dαpreserves small products.
As a similar result of Proposition 2.10, we have:
2.12 Proposition. Let Dbe a right exact I-diagram of abelian categories. If Disatisfies the axiom
AB3∗(resp., AB4∗)for any i∈Ob(I), then so does D-Rep.
2.3. Right representations over diagrams of categories. In this subsection we describe right
representations over diagrams of categories as well as their elementary properties. Since these
results are dual to the corresponding ones of left representations, we omit all proofs.
2.13 Definition. Let (D, η, τ ) be an I-diagram of categories. A right representations Nover D
consists of the following data:
•for i∈Ob(I), an object Niin Di, and
•for α:i→jin Mor(I), a structural morphism Nα:Nj→Dα(Ni) in Dj
such that the following two axioms are satisfied.
(rRep.1) Given composable morphisms iα
→jβ
→k∈Mor(I), there exists a commutative diagram
Nk
Nβα
//
Nβ
Dβα(Ni)
Dβ(Nj)
Dβ(Nα)
//Dβ(Dα(Ni))
τβ,α (Ni)
OO
in Dk, that is, Nβα =τβ,α(Ni)◦Dβ(Nα)◦Nβ.
(rRep.2) Given an object i∈Ob(I), there exists a commutative diagram
Ni
idNi//
Nei##
●
●
●
●
●
●
●
●
●Ni
Dei(Ni)
η−1
i(Ni)
;;
✇
✇
✇
✇
✇
✇
✇
✇
✇
in Di, that is, Nei=ηi(Ni).
A morphism σ:N→N′between two right representations Nand N′over Dis a family
{σi:Ni→N′
i}i∈Ob(I)of morphisms such that the following diagram
Nj
σj
//
Nα
N′
j
N′
α
Dα(Ni)Dα(σi)
//Dα(N′
i)
in Djcommutes for any α:i→j∈Mor(I).
Denote by Rep-Dthe category of right representations over D, which is clearly additive when D
is an I-diagram of abelian categories.
A dual result of Theorem 2.6 is:
2.14 Theorem. Let Dbe a left exact I-diagram of abelian categories. Then Rep-Dis an abelian
category. Moreover, a sequence M→N→Kin Rep-Dis exact if and only if Mi→Ni→Kiis
exact in Difor every i∈Ob(I).
One can modify the constructions in Subsection 2.2 to obtain colimits and limits in Rep-D. In
particular, we have the following results.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 21
2.15 Proposition. Let Dbe a left exact I-diagram of abelian categories such that Dαpreserves
small products for any α∈Mor(I). If Disatisfies the axiom AB3∗(resp., AB4∗)for any i∈Ob(I),
then so does Rep-D.
2.16 Proposition. Let Dbe a left exact I-diagram of abelian categories. If Disatisfies the axiom
AB3 (resp., AB4,AB5)for any i∈Ob(I), then so does Rep-D.
We end this section by giving an explicit relation between left representations and right repre-
sentations.
2.17 Remark. Suppose that the I-diagram Dof abelian categories admits enough right adjoints.
By Proposition 1.5, Dinduces an Iop -diagram Lof abelian categories admitting enough left ad-
joints. Furthermore, for each α∈Mor(I), the functor Lαop is nothing but the right adjoint of Dα
by definition. One then concludes that D-Rep can be identified with Rep-L. Indeed, given an
object Min D-Rep and a morphism α∈Mor(I), let M∗
αbe the adjoint morphism of Mα. Then by
Yoneda’s Lemma, Mis a left representation over Dif and only if the following hold:
•for i∈Ob(I), an object Miin Li=Di, and
•for αop :j→iin Mor(Iop), a structural morphism M∗
α:Mi→Lαop (Mj)
such that the axioms (rRep.1) and (rRep.2) in Definition 2.13 are satisfied, that is, Mis an ob-
ject in Rep-L. Furthermore, one can check that the above construction also gives a bijective
correspondence between morphisms in D-Rep and morphisms in Rep-L.
On the other hand, if the I-diagram Dof abelian categories admitting enough left adjoints,
then by Proposition 1.5, Dinduces an Iop-diagram Kof abelian categories admitting enough right
adjoints. Analogously, the categories Rep-Dand K-Rep coincide.
3. Functors induced by morphisms between diagrams
In this section we investigate the extension and the restriction of scalars along a given morphism
between two I-diagrams of abelian categories, and show that under some mild assumptions they
form an adjoint pair. Furthermore, we prove that a “locally identical” morphism between diagrams
of abelian categories induces an isomorphism between categories of representations.
Some results in this section have been obtained by H¨uttemann and R¨ondigs for the special case
of diagrams admitting enough right adjoints; for details, please refer to [44].
Let F: (D′, η′, τ ′)→(D, η, τ ) be a morphism between two I-diagrams of abelian categories. We
define an induced functor F!:D′-Rep →D-Rep as follows: given an object Nand a morphism
ω={ωi:Ni→N′
i}i∈Ob(I):N→N′in D′-Rep,
•for i∈Ob(I), F!(N)iis Fi(Ni);
•for α:i→jin Mor(I), F!(N)α:Dα(F!(N)i) = Dα(Fi(Ni)) →F!(N)j=Fj(Nj) is the
composite Dα(Fi(Ni)) F−1
α(Ni)
//Fj(D′
α(Ni)) Fj(Nα)
//Fj(Nj);
•F!(ω) : F!(N)→F!(N′) is defined to be {Fi(ωi) : F!(N)i=Fi(Ni)→Fi(N′
i) = F!(N′)i}i∈Ob(I).
We have the following result, which can be viewed as an extension along the morphism between
the two I-diagrams. For I-diagrams admitting enough right adjoints, this result has been established
in [44, Lemma 2.3.5].
3.1 Proposition. The above construction gives a functor F!:D′-Rep →D-Rep.
Proof. We prove that F!is a functor by checking that F!(N) is an object in D-Rep and F!(ω) is
a morphism in D-Rep. Other axioms of functors can be verified similarly. A detailed verification
can be found in Subsection B.3.
22 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
A dual construction F!:Rep-D′→Rep-D7for right representations is as follows: given an object
Nand a morphism σ={σi:Ni→N′
i}i∈Ob(I):N→N′in Rep-D′,F!(N) is defined by:
•for i∈Ob(I), F!(N)iis Fi(Ni);
•for α:i→j∈Mor(I), F!(N)α:F!(N)j=Fj(Nj)→Dα(F!(N)i) = Dα(Fi(Ni)) is the
composite Fj(Nj)Fj(Nα)
//Fj(D′
α(Ni)) Fα(Ni)
//Dα(Fi(Ni));
•F!(σ) : F!(N)→F!(N′) is defined to be {Fi(σi) : F!(N)i=Fi(Ni)→Fi(N′
i) = F!(N′)i}i∈Ob(I).
A dual version of Proposition 3.1 for right representations is:
3.2 Proposition. The above construction gives a functor F!:Rep-D′→Rep-D.
Under certain assumptions, the right adjoint of F!:D′-Rep →D-Rep exists. This fact has been
established in [44, Corollary 2.3.7], where a detailed proof is not given. For the convenience of the
reader, we give a complete proof.
3.3 Proposition. Let F:D′→Dbe a morphism of I-diagrams of abelian categories such that
both D′and Dadmit enough right adjoints and Fiadmits a right adjoint for any i∈Ob(I). Then
Finduces a functor F∗:D-Rep →D′-Rep, which is the right adjoint of F!.
Proof. By Proposition 1.8, Finduces a morphism G:L→L′, where Land L′are the Iop -
diagrams of abelian categories induced by Dand D′respectively. By Proposition 3.2, there exists
a functor
G!:Rep-L→Rep-L′,
which is the extension of scalars along G. However, according to Remark 2.17, the categories Rep-L
and D-Rep coincide, and the categories Rep-L′and D′-Rep coincide as well. In this way we obtain
the desired functor F∗which is defined to be G!.
Let Mbe an object in D-Rep and Nan object in D′-Rep. We prove the adjunction by explicitly
constructing a pair of natural maps
u: HomD′-Rep(N , F ∗(M)) ⇄HomD-Rep(F!(N), M ) : v,
which are inverse to each other. Note that for any i∈Ob(I), (Fi, Gi) is an adjoint pair by the
definition of G(see the proof of Proposition 1.8). Taking a morphism
ω={ωi:Ni→F∗(M)i=G!(M)i=Gi(Mi)}i∈Ob(I)
from Nto F∗(M), we set u(ω) to be
{u(ω)i:F!(N)i=Fi(Ni)→Mi}i∈Ob(I),
where u(ω)iis the adjoint morphism of ωiwith respect to the adjoint pair (Fi, Gi). On the other
hand, for a morphism
σ={σi:F!(N)i=Fi(Ni)→Mi}i∈Ob(I)
from F!(N) to M, we set v(σ) to be
{v(σ)i:Ni→Gi(Mi) = G!(M)i=F∗(M)i}i∈Ob(I),
where v(σ)iis the adjoint morphism of σiwith respect to the adjoint pair (Fi, Gi).
Now the reader can check that u(ω) (resp., v(σ)) is a morphism in D′-Rep (resp., D-Rep), and
uand vare inverse to each other. Furthermore, the construction of uand vare natural in both
variants.
The dual result for right representations is:
7Here we use the same symbol F!for both the left representation category and the right representation category.
We believe that it will not cause a big trouble to the reader since he/she can easily distinguish the two cases from
the context.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 23
3.4 Proposition. Let F:D′→Dbe a morphism of I-diagrams of abelian categories such that
both D′and Dadmit enough left adjoints and Fiadmits a left adjoint for every i∈Ob(I). Then
Finduces a functor F∗:Rep-D→Rep-D′, which is the left adjoint of F!.
The next result asserts that a “locally identical” morphism between two I-diagrams of abelian
categories induces an isomorphism between their representation categories.
3.5 Proposition. Let F: (D′, η′, τ ′)→(D, η, τ )be a morphism between I-diagrams of abelian
categories such that Fiis the identity for every i∈Ob(I). Then Finduces an isomorphism between
D′-Rep and D-Rep (resp., Rep-D′and Rep-D).
Proof. We only show the conclusion for the categories of left representations. By definition, there
exists a family {Fα:D′
α→Dα}α∈Mor(I)of natural isomorphisms satisfying the axioms (Mor.1) and
(Mor.2) in Definition 1.6. We prove the conclusion by explicitly constructing a pair of functors
U:D′-Rep ⇄D-Rep :V,
which are inverse to each other.
Let M′be an object in D′-Rep. Define U(M′) as follows:
•for i∈Ob(I), set U(M′)i=M′
i;
•for α:i→j∈Mor(I), set
U(M′)α=M′
α◦F−1
α(M′
i) : Dα(U(M′)i)→U(M′)j.
Let δ′:M′→K′be a morphism in D′-Rep, that is, δ′is a family {δ′
i:M′
i→K′
i}i∈Ob(I)of
morphisms such that
δ′
j◦M′
α=K′
α◦D′
α(δ′
i) (†)
for any morphism α:i→j∈Mor(I). Set U(δ′) to be δ′={δ′
i:M′
i→K′
i}i∈Ob(I).
We claim that U(M′) is a representation over Dand U(δ′) is a morphism. A detailed proof
of this claim can be found in Subsection B.4. Furthermore, one can check that Usatisfies other
axioms of functors.
Analogously, one can define the functor Vby sending an object Min D-Rep to the object V(M)
with V(M)i=Mifor any object i∈Ob(I) and V(M)α=Mα◦Fα(Ms(α)) for any morphism
α∈Mor(I), and sending a morphism δ:M→Kin D-Rep to δitself. Again, one can check that
Vis indeed a functor, and Uand Vare inverse to each other. This completes the proof.
As an immediate consequence of Propositions 1.11 and 3.5, the following result asserts that
whenever studying representations over a diagram Dof abelian categories indexed by a free category
associated to a quiver, without loss of generality one can suppose that Dis strict. This fact will
be frequently used in Part III to significantly simplifies many arguments.
3.6 Corollary. Let Ibe the free category associated to a quiver and Dan I-diagram of abelian
categories. Then there is a strict I-diagram D′such that the categories D-Rep and D′-Rep (resp.,
Rep-Dand Rep-D′) are isomorphic.
3.7 Remark. Let Ibe the free category associated to a quiver Q= (Q0, Q1) and Da strict I-
diagram of abelian categories. Since a non-identity morphism αin Ican be written uniquely as
α=αl◦ · · · ◦ α1with all α1,··· , αlarrows in Q1, each object Min D-Rep, up to isomorphism, is
uniquely determined by the family {Mi∈Di}i∈Q0of objects and the family {Mθ}θ∈Q1of structural
morphisms. In particular,
Mα=(Mei= idMi,if α=ei;
Mαl◦Dαl(Mαl−1)◦ · · · ◦ (Dαl◦ · · · ◦ Dα2)(Mα1),otherwise.
24 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Moreover, given two objects Mand M′in D-Rep, a morphism ω:M→M′in D-Rep is determined
by a family {ωi:Mi→M′
i}i∈Ob(I)of morphisms such that the diagram
Dα(Mi)Dα(ωi)
//
Mα
Dα(M′
i)
M′
α
Mj
ωj
//M′
j
in Djcommutes for all arrows α:i→jin Q1.
Similarly, an object Nin Rep-Dis completely determined by the family {Ni∈Di}i∈Q0of objects
and the family {Nα}α∈Q1of structural morphisms. Accordingly, a morphism σ:N→N′in Mod-D
is completely determined by a family {σi:Ni→N′
i}i∈Ob(I)of morphisms such that the diagram
Nj
σj
//
Nα
N′
j
N′
α
Dα(Ni)Dα(σi)
//Dα(N′
i)
in Djcommutes for all arrows α:i→jin Q1.
4. Representations over diagrams of module categories
Estrada and Virili introduced in [26] a representation Rof small categories Ion Cat (the 2-category
of small preadditive categories) and modules over R, and established a few fundamental homological
facts on the category of these modules. We observe that a representation Rof Iin that paper
is precisely an I-diagram of small preadditive categories considered in this paper. Therefore, it
is natural to explore relations between modules over Rin their sense and representations over
diagrams in our sense.
In this section, we show that an I-diagram Rof associative rings (or a representation Rof Ion
Ring in [26]) induces a right exact and endo-trivial I-diagram Rof module categories such that
the category R-Mod of left R-modules in [26] coincides with the category R-Rep. Furthermore,
all results on I-diagrams of associative rings in this section can extend to I-diagrams of small
preadditive categories (or a representation of Ion Cat in [26]); see Appendix A. Therefore, the
work described in [26] also falls into our framework.
4.1. Diagrams of module categories. Now we explain that an I-diagram Rof associative rings
induces a right exact and endo-trivial I-diagram Rof module categories. Explicitly,
•for i∈Ob(I), Riis a unitary associative ring, and
•for α:i→j∈Mor(I), Rα:Ri→Rjis a ring homomorphism.
Define Ras follows:
•for i∈Ob(I), set Rito be Ri-Mod, the category of left Ri-modules;
•for α:i→j∈Mor(I), set Rα:Ri-Mod →Rj-Mod to be the tensor product functor
Rj⊗Ri−;
•for i∈Ob(I), set ηi: idRi-Mod →Rei=Ri⊗Ri−to be the classical natural isomorphism
defined by the isomorphism M∼
=
−→ Ri⊗RiMfor any left Ri-module M;
•for a pair of composable morphisms α:i→jand β:j→kin Mor(I), set τβ,α :Rβ◦Rα→
Rβα to be the composite of the following classical natural isomorphisms
Rk⊗Rj(Rj⊗Ri−)∼
=
−→ (Rk⊗RjRj)⊗Ri−∼
=
−→ Rk⊗Ri−.
Dually, define Ras follows:
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 25
•for i∈Ob(I), set Rito be Mod-Ri, the category of right Ri-modules;
•for α:i→j∈Mor(I), set Rα:Mod-Ri→Mod-Rjto be the Hom functor HomRi(Rj,−);
•for i∈Ob(I), set ηi: idMod-Ri→Rei= HomRi(Ri,−) to be the classical natural isomor-
phism defined by the isomorphism N∼
=
−→ HomRi(Ri, N ) for any right Ri-module N;
•for a pair of composable morphisms α:i→jand β:j→kin Mor(I), set τβ,α :Rβ◦Rα→
Rβα to be the composite of the following classical natural isomorphisms
HomRj(Rk,HomRi(Rj,−)) ∼
=
−→ HomRi(Rk⊗RjRj,−)∼
=
−→ HomRi(Rk,−).
It is easy to show that both Rand Rare I-diagrams of abelian categories. Moreover, the following
facts are also clear from the above construction: Ris right exact and admits enough right adjoints;
Ris left exact and admits enough left adjoints, and both Rand Rare endo-trivial.
Recall from [26, Definition 3.5] that Ris said to be flat if Rαis a flat ring homomorphism for
any α:i→jin Mor(I), that is, the functor Rj⊗Ri−:Ri-Mod →Rj-Mod is exact. The following
example is taken from [26, Example 3.2], which contains three important algebraic structures arising
in geometric context. All these algebraic structures can be naturally viewed as flat diagrams of
associative rings indexed by the corresponding skeletal small categories.
4.1 Example. (1) Let (X, OX) be a scheme. Choose an affine open cover Uof Xand let Ibe the
small category associated to the poset Uordered by reverse inclusion. Then the structure sheaf
OXof rings is a flat I-diagram of commutative rings.
(2) Let Xbe an algebraic stack with the structure sheaf of rings OX, and denote by Ithe small
skeleton of the category of affine schemes smooth over X. Then the structure sheaf of rings OXis
a flat I-diagram of commutative rings.
(3) Let Xbe a Deligne-Mumford stack with the structure sheaf of rings OX, and let Ibe the
small skeleton of the category of affine schemes that are ´etale over X(such a small skeleton must
exist as ´etale morphisms are of finite type). Then the structure sheaf of rings OXis a flat I-diagram
of commutative rings.
Each I-diagram R=OXof commutative rings in Example 4.1 induces an I-diagram Rof module
categories. In particular, all three induced I-diagrams are exact.
The following result shall be well known to experts, which asserts that a morphism between two
presheaves of associative rings induces a functor between the corresponding categories of presheaves
of modules over them.
4.2 Proposition. Let F:R′→Rbe a morphism between I-diagrams of associative rings. Then
Finduces morphisms F:R′→Rand F:R′→Rbetween I-diagrams of module categories.
Proof. Note that Fi:R′
i→Riis a ring homomorphism for any i∈Ob(I), and both R′
α:R′
i→R′
j
and Rα:Ri→Rjare also ring homomorphisms for any α:i→j∈Mor(I). Define Fas follows:
•for i∈Ob(I), set the functor Fi:R′i=R′
i-Mod →Ri=Ri-Mod to be Ri⊗R′
i−;
•for α:i→j∈Mor(I), since R′αand Rαare R′
j⊗R′
i−and Rj⊗Ri−respectively, set
Fα:Fj◦R′α=Rj⊗R′
j(R′
j⊗R′
i−)→Rα◦Fi=Rj⊗Ri(Ri⊗R′
i−)
to be the composite of the following classical natural isomorphisms
Rj⊗R′
j(R′
j⊗R′
i−)∼
=
−→ (Rj⊗R′
jR′
j)⊗R′
i−∼
=
−→ Rj⊗R′
i−∼
=
−→ (Rj⊗RiRi)⊗R′
i−∼
=
−→ Rj⊗Ri(Ri⊗R′
i−).
It is not difficult to verify that Fsatisfies the axioms (Mor.1) and (Mor.2) in Definition 1.6. Thus,
Fis indeed a morphism from R′to R.
On the other hand, define Fas follows:
•for i∈Ob(I), set the functor Fi:R′i=Mod-R′
i→Ri=Mod-Rito be HomR′
i(Ri,−);
26 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
•for α:i→j∈Mor(I), since R′αand Rαare HomR′
i(R′
j,−) and HomRi(Rj,−) respectively,
set Fα:Fj◦R′α= HomR′
j(Rj,HomR′
i(R′
j,−)) →Rα◦Fi= HomRi(Rj,HomR′
i(Ri,−)) to
be the composite of the following classical natural isomorphisms
HomR′
j(Rj,HomR′
i(R′
j,−)) ∼
=
−→ HomR′
i(Rj⊗R′
jR′
j,−)∼
=
−→ HomR′
i(Rj,−)∼
=
−→
HomR′
i(Rj⊗RiRi,−)∼
=
−→ HomRi(Rj,HomR′
i(Ri,−)).
It is not difficult to verify that Fsatisfies the axioms (Mor.1) and (Mor.2) in Definition 1.6. Thus,
Fis a morphism from R′to R.
4.2. Representations over diagrams of module categories. As before, let (R, η, τ ) be an I-
diagram of associative rings. In this subsection we show that the category R-Mod of left R-modules
studied in [26] coincides with the category R-Rep8.
Let F1, F2:S→Tand H1, H2:Top →Ube four functors between categories, and let α:F1→F2
and β:H2→H1be two natural transformations. Recall that the Godement product of αand β
is the natural transformation β∗α:H2◦F2→H1◦F1by defining (β∗α)(S) as
(4.2.1) H1(α(S)) ◦β(F2(S)) = β(F1(S)) ◦H2(α(S))
for any object Sin S.
4.3 Definition (Estrada and Virili). Recall from [26, Definition 3.6] that a left R-module M
consists of the following data:
•for i∈Ob(I), a left Ri-module Mi, and
•for α:i→j∈Mor(I), a morphism Mα:Mi→(Rα)∗(Mj) in Ri-Mod, where (Rα)∗=
HomRj(Rj,−) is the restriction of scalars along Rα,
such that the following axioms are satisfied:
(Mod.1) Given composable morphisms iα
→jβ
→k∈Mor(I), there is a commutative diagram
Mi
Mβα
//
Mα
(Rβα)∗(Mk)
idMk∗τβ,α
(Rα)∗(Mj)(Rα)∗(Mβ)
//(Rα)∗((Rβ)∗(Mk)).
(Mod.2) For i∈Ob(I), there exists a commutative diagram
Mi
idMi//
Mei%%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑Mi
(Rei)∗(Mi)
idMi∗ηi
99
t
t
t
t
t
t
t
t
t
t
4.4 Remark. We mention that (Rβα)∗(Mk) = Mk◦Rβα and (Rα)∗((Rβ)∗(Mk)) = Mk◦Rβ◦Rαin
(Mod.1); one refers to (4.2.1) for the definition of idMk∗τβ,α . Also, note that (Rei)∗(Mi) = Mi◦Rei
in (Mod.2).
8The objects in R-Rep are called modules over the diagram Rby Greenlees and Shipley in [36].
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 27
A morphism ω:M→M′between two left R-modules is a family {ωi:Mi→M′
i}i∈Ob(I)of
morphisms such that the diagram
Mi
ωi//
Mα
M′
i
M′
α
(Rα)∗(Mj)(Rα)∗(ωj)
//(Rα)∗(M′
j)
in Ri-Mod commutes for any α:i→j∈Mor(I). Denote by R-Mod the category of left R-modules.
4.5 Proposition. Let Rbe an I-diagram of associative rings. Then the category R-Mod of left
R-modules coincides with the category R-Rep.
Proof. For any α:i→j∈Mor(I), note that ((Rα)!,(Rα)∗) is an adjoint pair, where (Rα)!=
Rj⊗Ri−. Let (Rα)!(Mi)→Mjbe the adjoint morphism of Mi→(Rα)∗(Mj). Then it is routine to
check by Yoneda’s Lemma that the definition of a left R-module Mis equivalent to the following
conditions:
•for i∈Ob(I), a left Ri-module Mi, and
•for α:i→j∈Mor(I), a homomorphism Mα: (Rα)!(Mi)→Mjin Rj-Mod
such that the following axioms are satisfied:
(Mod.1′) Given composable morphisms iα
→jβ
→k∈Mor(I), there is a commutative diagram
(Rβα)!(Mi)Mβα
//Mk
(Rβ)!((Rα)!(Mi)) (Rβ)!(Mα)
//
τβ,α (Mi)
OO
(Rβ)!(Mj).
Mβ
OO
(Mod.2′) For i∈Ob(I), there exists a commutative diagram
Mi
idMi//
ηi(Mi)%%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑Mi
(Rei)!(Mi).
Mei
99
s
s
s
s
s
s
s
s
s
s
Therefore, a left R-module Mis precisely an object in R-Rep. Furthermore, by Yoneda’s Lemma
a morphism between two left R-modules is precisely a morphism in R-Rep.
4.6 Remark. Since Radmits enough right adjoints, it induces an Iop-diagram Lof module cat-
egories by Proposition 1.5. Therfore, R-Mod =R-Rep =Rep-Lby Proposition 4.5 and Remark
2.17.
28 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Part II. Functors, adjunctions, and their applications
In this part we explore functors between representation categories induced by functors between
index categories. Explicitly, by a standard procedure, a functor G:J→Iof skeletal small
categories as well as an I-diagram Dof abelian categories give rise to an induced J-diagram D◦G
of abelian categories as well as a restriction functor from D-Rep to (D◦G)-Rep. Furthermore,
under suitable conditions we construct the left adjoint functor of the restriction functor.
Some special cases (for instances, Jis a full subcategory consisting of only one object) are of
particular interest to us, so we also study them in details and obtain a few important functors (such
as the stalk functor and the evaluation functor) and their adjoints, which are extensively applied
in this part to deduce some useful results on the structure of D-Rep.
Setup. Throughout this part suppose that (D, η, τ ) is right exact I-diagram of abelian categories
unless otherwise specified, so that the category D-Rep is abelian by Theorem 2.6, though some
results still hold without this assumption.
5. The induction functor and its applications
5.1. The restriction functor and its left adjoint. It is well known that continuous maps be-
tween topological spaces induce functors between categories of sheaves of sets (or groups, modules).
In this section we establish the same result for diagrams of abelian categories and representations
over them. We mention that H¨uttemann and R¨ondigs have considered this question for the special
case of diagrams admitting enough right adjoints and established a few results appearing in this
subsection; for details, please refer to [44, Subsection 2.4].
Given a functor G:J→Iof skeletal small categories, we can obtain an associated J-diagram of
abelian categories via a process of composing functors.
5.1 Lemma. Let G:J→Ibe a functor between skeletal small categories. Then D◦Gis a right
exact J-diagram with (D◦G)i=DG(i)for i∈Ob(J)and (D◦G)λ=DG(λ)for λ∈Mor(J).
Proof. Note that the I-diagram Dis a pseudo 2-functor from I, viewed naturally as a 2-category,
to the meta-2-category AB of abelian categories; see Remark 1.2. Then the composite D◦Gis
also a pseudo 2-functor from Jto AB, and hence, a J-diagram. Specifically, the J-diagram D◦G
consists of the following data:
•for i∈Ob(J), an abelian category (D◦G)i=DG(i);
•for λ:i→j∈Mor(J), a functor (D◦G)λ: (D◦G)i→(D◦G)j, which is the functor
DG(λ):DG(i)→DG(j);
•for i∈Ob(J), a natural isomorphism µi: id(D◦G)i→(D◦G)ei, which is the natural
isomorphism ηG(i): idDG(i)→DG(ei)=DeG(i);
•for a pair of composable morphisms λand κin Mor(J), a natural isomorphism
νκ,λ : (D◦G)κ◦(D◦G)λ→(D◦G)κλ,
which is the natural isomorphism
τG(κ),G(λ):DG(κ)◦DG(λ)→DG(κλ).
It is clear from the above descriptions that D◦Gis right exact since so is D.
Now we begin to construct various functors between D-Rep and (D◦G)-Rep. The first one is
the restriction functor G∗:D-Rep →(D◦G)-Rep defined as follows: given an object Mand a
morphism ω={ωi:Mi→M′
i}i∈Ob(I):M→M′in D-Rep,
•for j∈Ob(J), set G∗(M)jto be MG(j)in DG(j);
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 29
•for λ:i→j∈Mor(J), set the structural morphism G∗(M)λ: (D◦G)λ(G∗(M)i)→G∗(M)j
to be the morphism MG(λ):DG(λ)(MG(i))→MG(j);
•set G∗(ω) : G∗(M)→G∗(M′) to be {ωG(i):MG(i)→M′
G(i)}i∈Ob(J).
5.2 Proposition. The above construction gives an exact functor G∗:D-Rep →(D◦G)-Rep.
Proof. By a routine check, we conclude that G∗(M) satisfies the axioms (lRep.1) and (lRep.2) in
Definition 2.1, and hence, is an object in (D◦G)-Rep. It is also clear that G∗(ω) is a morphism in
(D◦G)-Rep. Furthermore, one can check that G∗satisfies other axioms of functors, and is exact
by Theorem 2.6.
Suppose that Disatisfies the axiom AB3 for every i∈Ob(I) and Dαpreserves small coproducts
for every α∈Mor(I). By Remark 2.8, each Dihas small colimits for i∈Ob(I) and each Dαpreserves
small colimits. We can construct a functor G!: (D◦G)-Rep →D-Rep in the inverse direction, called
the induction functor. A more conceptual construction of this functor is given in [44, Subsection
2.4], where it is called a twisted left Kan extension. To help the reader understand this abstract
and complicated construction, here we give a more explicit and constructive description, but do
not claim originality.
For this purpose, we introduce a special construction of colimits over morphism categories.
Explicitly, for i∈Ob(I), let G/i be the over category whose objects are morphisms in Istarting at
G(j) for a certain j∈Ob(J) and ending at i, and morphisms from θ:G(j)→ito θ′:G(j′)→i
are morphisms β:j→j′in Jsuch that θ=θ′G(β). Now given a representation N∈(D◦G)-Rep,
we define a functor ˜
N:G/i →Dias follows:
•for θ:G(j)→i, let ˜
N(θ) = Dθ(Nj);
•for a morphism β:j→j′from θ:G(j)→ito θ′:G(j′)→i, let ˜
N(β) be the composite
of the following morphisms Dθ(Nj) = Dθ′G(β)(Nj)∼
=Dθ′(DG(β)(Nj)) →Dθ′(Nj′) where the
last map is obtained by applying Dθ′to the structural map DG(β)(Nj)→Nj′.
It is easy to check that ˜
Nis indeed a functor, so one can define colimG/i ˜
N. Frequently we also
denote this colimit by colimG/i Dθ(N•) where θdenotes all morphisms G(•)→iwith • ∈ Ob(J).
Now we define the induction functor as follows. Given an object Nand a morphism σ:N→N′
in (D◦G)-Rep, we define:
•For i∈Ob(I), set G!(N)ito be colimG/i Dθ(N•).
•For α:i→j∈Mor(I), note that αθ ∈HomI(G(•), j) for any morphism θ∈HomI(G(•), i),
so there exists a canonical morphism
sαθ :Dαθ(N•)−→ colim
G/j
Dδ(N•).
By the universal property of colimits, we can find a unique morphism ϑαsuch that the
diagram
(5.2.1) Dα(Dθ(N•)) τα,θ(N•)
//
sα,θ
θ
Dαθ(N•)
sαθ
colimG/i Dα(Dθ(N•)) ϑα//colimG/j Dδ(N•)
30 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
commutes. On the other hand, by the universal property of colimits again, there exists a
morphism χαsuch that the diagram
(5.2.2) Dα(Dθ(N•))
Dα(sθ)
**
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
sα,θ
θ
colimG/i Dα(Dθ(N•)) χα//Dα(colimG/i Dθ(N•))
commutes. We mention that Dαpreserves small colimits (see Remark 2.8), that is, the
morphism χαis an isomorphism. Thus we set the structural morphism G!(N)αto be the
composite ϑα◦χ−1
α.
•For any object θ∈G/i, there exists a morphism Dθ(σ•) : Dθ(N•)→Dθ(N′
•). By the
universal property of colimits, we can find a unique morphism ωisuch that the diagram
(5.2.3) Dθ(N•)Dθ(σ•)
//
sθ
Dθ(N′
•)
s′
θ
colimG/i Dθ(N•) = G!(N)i
ωi//G!(N′)i= colimG/i Dθ(N′
•)
commutes. Define G!(σ) : G!(N)→G!(N′) to be {ωi}i∈Ob(I).
5.3 Example. Colimits appearing in the above construction seems mysterious, so let us give an
explicit example for illustration. Let Q= (Q0, Q1) be a quiver without loops (arrows from a vertex
to itself) and oriented cycles (that is, a nontrivial path which is of length at least two and starts
and ends at the same vertex), and let Q′= (Q′
0, Q′
1) be the subquiver obtained by removing from
Qa vertex jand all paths through it. Since each quiver can be viewed as a category in a natural
way, we obtain an inclusion functor ι:Q′→Q. Then objects in ι/j are paths γ:i→jsuch that
i6=j, and morphisms from γ:i→jto γ′:i′→jare paths β:i→i′such that γ=γ′β(here we
need the assumption that Qhas no loops or oriented cycles to guarantee that every path from ito
i′in Qis also contained in Q′). It is easy to deduce the following observations:
(1) ι/j has a poset structure given by γ6γ′if γ=γ′βfor a certain β;
(2) each connect component of ι/i contains a unique arrow γ:• → j, which is the terminal
object of this component.
By these observations, the colimit over ι/i becomes the coproduct indexed by arrows ending at j:
colim
ι/j
Dθ(N•) = a
θ∈Q1(•,j)
Dθ(N•).
The following result is a generalization of a well known fact in representation theory of rings
and sheaf theory on topological spaces: the restriction functor and the induction functor form an
adjoint pair, though its proof becomes much more complicated in our very general framework. We
shall point out that H¨uttemann and R¨ondigs have proved this result in [44, Theorem 2.4.1] for
I-diagrams admitting enough right adjoints. We do not make this assumption, and hence the result
can apply to a wider framework.
5.4 Theorem. Suppose that Disatisfies the axiom AB3 for i∈Ob(I)and Dαpreserves small
coproducts for α∈Mor(I). Then G!defined above is a functor, and is the left adjoint of G∗.
Proof. For the first statement, we prove that G!sends objects to objects and morphisms to mor-
phisms; the other axioms of functors are clear since the above construction is clearly functorial. To
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 31
show that G!(N) is an object in D-Rep, we verify the axioms (lRep.1) and (lRep.2) in Definition
2.1. Explicitly, we shall have:
G!(N)β◦Dβ(G!(N)α)◦τ−1
β,α(G!(N)i) = G!(N)βα (†)
for every pair of composable morphisms α:i→jand β:j→kin Mor(I), and
G!(N)ei◦ηi(G!(N)i)◦sθ=sθ(‡)
for any i∈Ob(I). Similarly, to verify that {ωi}i∈Ob(I)is a morphism in D-Rep, we shall have
ωj◦G!(N)α=G!(N′)α◦Dα(ωi) (♯)
for any α:i→jin Mor(I). Detailed proofs of these identities are given in Subsection B.5.
Now we prove the second statement by explicitly constructing a pair of natural maps for every
object Nin (D◦G)-Rep and every object Min D-Rep
u: HomD-Rep(G!(N), M )⇄Hom(D◦G)-Rep(N, G∗(M)) : v,
and showing that uand vare inverse to each other.
To define the map u, take a morphism ω={ωi}i∈Ob(I):G!(N)→Min D-Rep. For i∈Ob(J),
note that Ni∈(D◦G)i=DG(i)and µi=ηG(i)(see the proof of Lemma 5.1). Define u(ω)ito be
the composite
Ni
ηG(i)(Ni)
−→ DeG(i)(Ni)
seG(i)
−→ G!(N)G(i)= colim
G/G(i)
Dθ(N•)ωG(i)
−→ MG(i)=G∗(M)i,
that is, u(ω)i=ωG(i)◦seG(i)◦ηG(i)(Ni). Then for any morphism λ:i→j∈Mor(J), we have
G∗(M)λ◦(D◦G)λ(u(ω)i) = u(ω)j◦Nλ,(♮)
whose proof is given in Subsection B.5. Thus {u(ω)i}i∈Ob(J)is a morphism in (D◦G)-Rep. Now
define u(ω) to be {u(ω)i}i∈Ob(J).
To define the map v, let σ={σi}i∈Ob(J):N→G∗(M) be a morphism in (D◦G)-Rep. For
i∈Ob(I) and θ∈Ob(G/i), note that the morphism σ•:N•→G∗(M)•induces the morphism
Dθ(σ•) : Dθ(N•)→Dθ(G∗(M)•). By the universal property of colimits, there exists a unique
morphism v(σ)isuch that the diagram
(5.4.1) Dθ(N•)Dθ(σ•)
//
sθ
Dθ(G∗(M)•) = Dθ(MG(•))
Mθ
G!(N)i= colimG/i Dθ(N•)v(σ)i//Mi
commutes. Define v(σ) to be {v(σ)i}i∈Ob(I). Then for any morphism α:i→j∈Mor(I), we have
v(σ)j◦G!(N)α◦χα◦sα,θ
θ=Mα◦Dα(v(σ)i)◦χα◦sα,θ
θ(§)
whose proof is given in Subsection B.5. Since χαis an isomorphism, by the universal property of
colimits one has
v(σ)j◦G!(N)α=Mα◦Dα(v(σ)i),
and so v(σ) is a morphism in D-Rep.
32 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Clearly, the maps uand vconstructed above are natural with respect to Nand M. We prove
that they are inverse to each other. On one hand, for i∈Ob(J), one has
u(v(σ))i=v(σ)G(i)◦seG(i)◦ηG(i)(Ni) by def. of u(v(σ))i
=MeG(i)◦DeG(i)(σi)◦ηG(i)(Ni) by (5.4.1)
=η−1
G(i)(MG(i))◦DeG(i)(σi)◦ηG(i)(Ni) by (lRep.2)
=σi◦η−1
G(i)(Ni)◦ηG(i)(Ni)
=σi,
where the fourth equality holds by the commutative diagram
DeG(i)(Ni)
η−1
G(i)(Ni)
DeG(i)(σi)
//DeG(i)(MG(i))
η−1
G(i)(MG(i))
Ni
σi//MG(i)
induced by applying the natural isomorphism η−1
G(i)to the morphism σi. Thus, u(v(σ)) = σ, so uv
is the identity map.
On the other hand, for i∈Ob(I), we have
v(u(ω))i◦sθ=Mθ◦Dθ(u(ω)•) by (5.4.1)
=Mθ◦Dθ(ωG(•))◦Dθ(seG(•))◦Dθ(ηG(•)(N•)) by def. of u(ω)•
=ωi◦G!(N)θ◦Dθ(seG(•))◦Dθ(ηG(•)(N•))
=ωi◦ϑθ◦χ−1
θ◦Dθ(seG(•))◦Dθ(ηG(•)(N•)) by def. of G!(N)θ
=ωi◦ϑθ◦χ−1
θ◦χθ◦sθ,eG(•)
eG(•)◦Dθ(ηG(•)(N•)) by (5.2.2)
=ωi◦ϑθ◦sθ,eG(•)
eG(•)◦Dθ(ηG(•)(N•))
=ωi◦ϑθ◦sθ,eG(•)
eG(•)◦τ−1
θ,eG(•)(N•) by (Dia.2)
=ωi◦sθby (5.2.1).
Therefore, by the universal property of colimits, we have v(u(ω))i=ωiand hence v(u(ω)) = ω.
Consequently, vu is also the identity map.
5.5 Example. The above construction unifies many constructions familiar to the reader. For
instances, if Iis a group (a category with one object such that all endomorphisms are invertible),
and Iis a subgroup. Take Dto be the trivial diagram over a module category. Then we recover
the classical induction functor in group representation theory. Similarly, the extension functor
along a ring homomorphism is also a special case of this general construction. More generally, this
construction also includes the classical left Kan extension in category theory.
5.6 Remark. When the diagram Dadmits enough right adjoints and each Diis complete, one can
construct the right adjoint functor of G∗, using the right adjoint of Dαfor α∈Mor(I). However,
for an arbitrary diagram D, we do not know whether its right adjoint exists.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 33
Proposition 3.3 and the above theorem are two distinct aspects of a general base change proce-
dure. Explicitly, given the following diagram which might not be commutative
JG//
D′
❅
❅
❅
❅
❅
❅
❅
❅I
D
~~⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
AB
with D′a right exact J-diagram of abelian categories. If both D′and Dadmit enough right adjoints,
and there is a morphism F:D′→D◦Gsuch that Fjadmits a right adjoint for j∈Ob(J), then
by Proposition 3.3 one obtains an adjoint pair of functors
F!:D′-Rep ⇄(D◦G)-Rep :F∗.
On the other hand, if Disatisfies the axiom AB3 for i∈Ob(I) and Dαpreserves small coproducts
for α∈Mor(I), then by Theorem 5.4 one obtains another adjoint pair of functors
G!: (D◦G)-Rep ⇄D-Rep :G∗.
Combining these two facts, we get the following result.
5.7 Corollary. Suppose that the conditions specified in Proposition 3.3 and Theorem 5.4 hold.
Then G!◦F!:D′-Rep ⇄D-Rep :F∗◦G∗is an adjoint pair.
Now we apply the general results to a special case. Take an object i∈Ob(I). Let Iibe the
subcategory of Iconsisting of the single object iand the single morphism ei. Note that there exists
an obvious isomorphism between (D◦ιi)-Rep and Di, where ιi:Ii→Iis the canonical inclusion.
Explicitly, an object Miin Dican be viewed as an object in (D◦ιi)-Rep with the structural morphism
(Mi)ei: (D◦ιi)ei(Mi) = Dei(Mi)→Mi, which is exactly η−1
i(Mi). A morphism ωi:Mi→M′
iin
Digives a commutative diagram
(D◦ιi)ei(Mi) = Dei(Mi)
Dei(ωi)
//
(Mi)ei
(D◦ιi)ei(M′
i) = Dei(M′
i)
(M′
i)ei
Mi
ωi//M′
i
and hence it is also a morphism in (D◦ιi)-Rep. Therefore, in the rest of this paper we identify
these two categories.
We define the evaluation functor at ito be the composite
evai:D-Rep (ιi)∗
−→ (D◦ιi)-Rep ≃
−→ Di
sending a left representation Mover Dto its “local” value Miin Di. If Djsatisfies the axiom AB3
for every j∈Ob(I) and Dαpreserves small coproducts for every α∈Mor(I), then by Theorem 5.4,
the inclusion ιi:Ii→Iinduces a functor
frei:Di≃
−→ (D◦ιi)-Rep (ιi)!
−→ D-Rep.
The functor freihas a very simple description. Indeed, since Iicontains only one morphism, the
over category ιi/j for any object j∈Ob(I) is discrete. Therefore, all colimits appearing in that
construction become coproducts. That is, for Mi∈Diand j∈Ob(I), one has
(frei(Mi))j=a
θ∈HomI(i,j)
Dθ(Mi).
An immediate corollary of Theorem 5.4 is:
34 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
5.8 Corollary. Let ibe an object in Ob(I). Suppose that Djsatisfies the axiom AB3 for j∈Ob(I)
and Dαpreserves small coproducts for α∈Mor(I). Then (frei,evai)is an adjoint pair.
5.2. Grothendieck structure and locally finitely presented property. The evaluation func-
tor and its left adjoint can serve as a bridge connecting the global datum of D-Rep and local data of
Di’s. In this subsection we illustrate some applications, and establish a few fundamental results on
the structure of D-Rep. In particular, local Grothendieck structures on Di’s can amalgamate to a
Grothendieck structure on D-Rep, and under certain conditions, D-Rep is locally finitely presented.
Consequently, we can obtain a generalization of the classical representation theorem of Makkai and
Par´e [50] (see also [1] and [12]).
5.9 Proposition. Suppose that Disatisfies the axiom AB3 for i∈Ob(I)and Dαpreserves small
coproducts for α∈Mor(I). If Dihas a family of generators (resp., projective generators) for every
i∈Ob(I), then so does D-Rep.
Proof. Denote by Githe family of generators of Difor i∈Ob(I). We claim that
fre(G) = {frei(Gi)|Gi∈Gi, i ∈Ob(I)}
is a family of generators of D-Rep.
Let ω:M→M′be a non-zero morphism in D-Rep. Then there exists an object i∈Ob(I) such
that ωi:Mi→M′
iis non-zero. Since Giis a family of generators of Di, one can find a morphism
gi:Gi→Miwith Gi∈Gisuch that ωi◦gi6= 0. We construct a morphism σ:frei(Gi)→M
such that ω◦σ6= 0, which implies the claim. It turns out that such a σcan be expressed as the
composite of two morphisms.
To construct the first morphism, applying the functor freito gi, we obtain a morphism frei(gi) :
frei(Gi)→frei(Mi) such that the diagram
(5.9.1) Dei(Gi)
Dei(gi)
//
εGi
ei
Dei(Mi)
εMi
ei
`ι∈HomI(i,i)Dι(Gi) = frei(Gi)i
frei(gi)i
//frei(Mi)i=`ι∈HomI(i,i)Dι(Mi)
in Dicommutes, where εGi
eiand εMi
eiare the canonical injections; see (5.2.3).
To construct the second morphism, for j∈Ob(I), by the universal property of coproducts, there
exists a unique morphism ρjsuch that for every θ∈HomI(i, j) the diagram
(5.9.2) Dθ(Mi)
εMi
θ
Mθ
))
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
`θ∈HomI(i,j)Dθ(Mi) = frei(Mi)jρj
//Mj
in Djcommutes. Note that Dαpreserves small coproducts for α∈Mor(I) by assumption. It is
routine to check that ρ={ρj}j∈Ob(I)is a morphism from frei(Mi) to M.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 35
Now let σ=ρ◦frei(gi). To verify ω◦σ6= 0, it is enough to show ωi◦σi6= 0. Indeed, we have
ωi◦σi◦εGi
ei◦ηi(Gi) = ωi◦ρi◦frei(gi)i◦εGi
ei◦ηi(Gi)
=ωi◦ρi◦εMi
ei◦Dei(gi)◦ηi(Gi) by (5.9.1)
=ωi◦Mei◦Dei(gi)◦ηi(Gi) by (5.9.2)
=ωi◦η−1
i(Mi)◦Dei(gi)◦ηi(Gi)
=ωi◦gi◦η−1
i(Gi)◦ηi(Gi)
=ωi◦gi6= 0,
where the fifth equality holds by the commutative diagram induced by applying the natural iso-
morphism η−1
ito the morphism gi.
In particular, if the objects in Giare projective for i∈Ob(I), then fre(G) is a family of projective
generators of D-Rep since frei, as the left adjoint of the exact functor evai, preserves projective
objects.
5.10 Remark. According to the proof of Proposition 5.9, any projective object in D-Rep is iso-
morphic to a direct summand of a direct sum of members in the family
{frei(Pi)|i∈Ob(I) and Piis projective in Di}.
As an immediate consequence of Propositions 2.10 and 5.9, we have:
5.11 Theorem. Suppose that Dαpreserves small coproducts for α∈Mor(I). If Diis a Grothendieck
category (resp., a Grothendieck category with a family of projective generators) for every i∈Ob(I),
then so is D-Rep.
Let Rbe an I-diagram of associative rings. Estrada and Viliri showed in [26, Theorem 3.18]
that the category R-Mod in Definition 4.3 is a Grothendieck category. If further Iis a poset,
then R-Mod has a projective generator. According to Proposition 4.5, R-Mod coincides with the
category R-Rep, where Ris the I-diagram of module categories induced by R. Applying the above
theorem to this special case, we obtain the following result, which improves [26, Theorem 3.18] by
dropping the unessential condition that Iis a poset.
5.12 Corollary. Let Rbe an I-diagram of associative rings. Then R-Rep is a Grothendieck
category with a projective generator.
Let Abe an abelian category satisfying the axiom AB3. Recall that an object Xin Ais said to
be finitely presented provided that the representable functor HomA(X, −) commutes with filtered
colimits. Denote the full subcategory of finitely presented objects by Fp(A). Recall that Ais
locally finitely presented if Fp(A) is skeletally small and every object in Ais a filtered colimit of
finitely presented objects, or equivalently, Apossesses a family of finitely presented generators; see
[1, Theorem 1.11]9. The following proposition tells us that the locally finitely presented property
of each abelian category Dican also be amalgamated to the locally finitely presented property of
D-Rep.
5.13 Proposition. Suppose that Dαpreserves small coproducts for α∈Mor(I). If Diis a locally
finitely presented for all i∈Ob(I), then so is D-Rep.
Proof. By Proposition 5.9, it is enough to show that freipreserves finitely presented object in Di
for i∈Ob(I). Let Nibe a finitely presented object in Diand (Mx, f yx) be a filtered direct system
9In an abelian category, any set of generators is a set of strong generators.
36 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
of objects in D-Rep. Then we have
HomD-Rep(frei(Ni),colim Mx)∼
=HomDi(Ni,evai(colim Mx))
= HomDi(Ni,colim Mx
i)
∼
=colim HomDi(Ni, Mx
i)
∼
=colim HomD-Rep(frei(Ni), M x),
where the first isomorphism holds by Corollary 5.8, the second equality holds since evai(colim Mx) =
(colim Mx)iand the filtered colimits in D-Rep are computed componentwise (see Subsection 2.2),
the third isomorphism holds as Niis a finitely presented object in Di, and the last isomorphism
holds by Corollary 5.8 again.
The next result, following from [12, Theorem 1.4(2)] and Proposition 5.13, gives the representa-
tion theorem for D-Rep. Recall from [52] that an object in the functor category Fun(Cop,Ab) (where
Cis a skeletally small additive category) is called flat if it is a colimit of representable functors.
5.14 Theorem. Suppose that Dαpreserves small coproducts for α∈Mor(I). If Diis locally
finitely presented for i∈Ob(I), then D-Rep is equivalent to the subcategory of flat objects in the
functor category Fun(Fp(D-Rep)op,Ab).
5.15 Remark. A similar result was given in [26, Theorem 4.13] for R-Rep, based on a specific
tensor product constructed in the representation category. However, we shall remind the reader
that flat objects defined in that paper are different from flat objects in our paper; see [26, Definition
3.20] for the precise meaning.
5.3. Dual results for Rep-D.In this subsection we describe the dual constructions and results
for the category Rep-D, where Dis assumed to be a left exact I-diagram of abelian categories. For
the convenience of the reader, we give brief constructions, but omit detailed proofs.
Let G:J→Ibe a functor between two skeletal small categories. The restriction functor
G∗:Rep-D→Rep-(D◦G) is defined as follows. Given an object Kin Rep-D, the object G∗(K)
consists of the following data:
•for i∈Ob(J), G∗(K)i=KG(i)in DG(i);
•for λ:i→j∈Mor(J), set the structural morphism G∗(K)λ:G∗(K)j→(D◦G)λ(G∗(K)i)
to be the morphism KG(λ):KG(j)→DG(λ)(KG(i)).
Given a morphism ω={ωi}i∈Ob(I):K→K′in Rep-D, the morphism G∗(ω) : G∗(K)→G∗(K′)
in Rep-(D◦G) is defined to be {ωG(i):KG(i)→K′
G(i)}i∈Ob(J).
Dual to Proposition 5.2, we have:
5.16 Proposition. The above construction G∗:Rep-D→Rep-(D◦G)is an exact functor.
Suppose that Disatisfies the axiom AB3∗for i∈Ob(I) and Dαpreserves small products for
α∈Mor(I). By Remark 2.8, Dihas small limits for i∈Ob(I) and Dαpreserves small limits for
α∈Mor(I). In this case, the induction functor G!:Rep-(D◦G)→Rep-Dcan be defined by a
construction dual to the one described in Subsection 5.1 via replacing all colimits by limits.
The following result is dual to Theorem 5.4. For the reader’s convenience, we give in the proof
the definitions of the two inverse maps, while leave the process of verifying the relevant axioms and
corresponding commutative diagrams to the interested reader.
5.17 Theorem. Let G:J→Ibe a functor between skeletal small categories. Suppose that Di
satisfies the axiom AB3∗for i∈Ob(I)and Dαpreserves small products for α∈Mor(I). Then G!
defined above is a functor, and is the right adjoint of G∗.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 37
Proof. Let Lbe an object in Rep-(D◦G) and Kan object in Rep-D. The definitions of the pair
of inverse maps
s: HomRep-D(K, G!(L)) ⇄HomRep-(D◦G)(G∗(K), L) : t
are given as follows.
Take a morphism ρ={ρi}i∈Ob(I):K→G!(L) in Rep-D. For j∈Ob(J), note that Lj∈
(D◦G)j=DG(j)and µj=ηG(j)(see the proof of Lemma 5.1). We have the following maps:
G∗(K)j=KG(j)
ρG(j)
−→ G!(L)G(j)= lim
G/G(j)
Dθ(L•)peG(j)
−→ DeG(j)(Lj)η−1
G(j)(Lj)
−→ Lj.
Define s(ρ)j=η−1
G(j)(Lj)◦peG(j)◦ρG(j)and s(ρ) to be {s(ρ)j}j∈Ob(J).
Let ς={ςj}j∈Ob(J):G∗(K)→Lbe a morphism in Rep-(D◦G). For i∈Ob(I) and θ∈Ob(G/i),
note that the morphism ς•:G∗(K)•→L•induces the morphism
Dθ(ς•) : Dθ(G∗(K)•)→Dθ(L•).
By the universal property of limits, there exists a unique morphism t(ς)isuch that the diagram
Ki
t(ς)i//
Kθ
G!(L)i= limG/i Dθ(L•)
pθ
Dθ(G∗(K)•) = Dθ(KG(•))Dθ(ς•)
//Dθ(L•)
commutes. Define t(ς) to be {t(ς)i}i∈Ob(I).
Now we describe the dual result of Corollary 5.7. Given the following diagram which might not
commute
JG//
D′
❅
❅
❅
❅
❅
❅
❅
❅I
D
~~⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
AB
with D′a left exact J-diagram of abelian categories. If both D′and Dadmit enough left adjoints,
and there is a morphism F:D′→D◦Gsuch that Fjadmits a left adjoint for j∈Ob(J), then by
Proposition 3.4 one obtains an adjoint pair of functors
F∗:Rep-(D◦G)⇄Rep-D′:F!.
On the other hand, if Disatisfies the axiom AB3∗for i∈Ob(I) and Dαpreserves small products
for α∈Mor(I), then by Theorem 5.17 one obtains another adjoint pair
G∗:Rep-D⇄Rep-(D◦G) : G!.
Combining these two facts, we have the following result.
5.18 Corollary. Suppose that the conditions specified in Proposition 3.4 and Theorem 5.17 hold.
Then F∗◦G∗:Rep-D⇄Rep-D′:G!◦F!is an adjoint pair.
Fix an object i∈Ob(I) and let Iibe the category with object iand one morphism ei. The
evaluation functor at iis defined as
evai:Rep-Dι∗
i
−→ Rep-(D◦ιi)≃
−→ Di.
Its left adjoint is
frei:Di≃
−→ Rep-(D◦ιi)(ιi)!
−→ Rep-D.
38 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
In particular, for Mi∈Diand j∈Ob(I), one has
(frei(Mi))j=Y
θ∈HomI(i,j)
Dθ(Mi).
As an immediate consequence of Theorem 5.17, we have:
5.19 Corollary. Let ibe an object in Ob(I). Suppose that Djsatisfies the axiom AB3∗for j∈
Ob(I)and Dαpreserves small products for α∈Mor(I). Then (evai,frei)is an adjoint pair.
A dual result of Proposition 5.9 asserts that under some assumptions Rep-Dhas enough injectives
provided that so does each Di.
5.20 Proposition. Suppose that Disatisfies the axiom AB3∗for i∈Ob(I)and Dαpreserves small
products for α∈Mor(I). If Dihas a family of cogenerators (resp., injective cogenerators) for
i∈Ob(I), then so does Rep-D.
Remark. Unfortunately, at this moment we do not know whether Rep-Dis a Grothendieck cate-
gory even with the assumption in the above proposition and the extra condition that each Diis an
Grothendieck category. However, if Dadmits enough left adjoints, then by Remark 2.17, Rep-D
can be identified with the left representation category over another diagram of abelian categories.
In this case, Rep-Dis an Grothendieck category under suitable conditions. In particular, when R
is an I-diagram of associative rings, both R-Rep and Rep-Rare Grothendieck categories.
6. The lift and stalk functors
We already know that a functor G:J→Ibetween two skeletal small categories as well as an
I-diagram Dof abelian categories induces a J-diagram D◦Gof abelian categories and a restriction
functor from D-Rep to (D◦G)-Rep. When J=I/Pis a subcategory of Iobtained by removing a
special subset Pof Mor(I) (called a prime ideal of Mor(I)) and its associated objects from I, we
can construct a functor in the inverse direction, from (D◦G)-Rep to D-Rep, called the lift functor.
Intuitively, this construction is similar to the following classical example: given a two-sided ideal
Iof an associative ring A, every A/I-module can be viewed as an A-module in a natural way,
and hence we obtain an embedding functor from the category of A/I-modules to the category of
A-modules, which is exactly the restriction functor along the quotient homomorphism A→A/I.
However, there are several remarkable differences: Pmust be a prime ideal rather than an arbitrary
ideal; I/Pis a subcategory (rather than a quotient category) of I; and the lift functor does not
coincide with the restriction functor along the embedding functor I/P→I.
In this section we systematically study this lift functor, and construct its left adjoint, the cokernel
functor. Since we are always interested to relate properties of D-Rep to corresponding properties
of Di(the local-global principle), we consider in details the case that I/Phas only one object. In
this special case we obtain the stalk functor and its left adjoint.
6.1. The lift functor and its left adoint. In this subsection, we introduce the lift functor lifP
and the cokernel functor cokPwith respect to a prime ideal Pof Mor(I), and show that they form
an adjoint pair.
A nonempty subset Pof Mor(I) is called a two-sided ideal of Iif for any α∈Mor(I) and any
morphism β∈P, one has αβ or βα, whenever composable, is always contained in P. The two-sided
ideal Pof Iis said to be prime if the converse statement holds, that is, one has α∈Por β∈P
whenever αβ ∈P.
From the above definition, it is easy to see that a two-sided ideal Pof Mor(I) is prime if and only
if Mor(I)\Pis closed under compositions of morphisms. More generally, ideals and prime ideals
can be defined for any nonempty set Sequipped with a “partial” binary operation, which is a map
from a subset of S×Sto S.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 39
Given a prime ideal P⊆Mor(I), we can construct a subcategory I/Pof Ias follows:
•i∈Ob(I) is also an object in Ob(I/P) if ei/∈P.
•α:i→j∈Mor(I) is contained in Mor(I/P) if α /∈P10.
Note that for any pair of composable morphisms αand βin Mor(I) such that neither αnor β
is in P, their composite βα is not in Psince Pis prime by assumption. Therefore, βα belongs
to Mor(I/P), so I/Pis indeed a subcategory. Furthermore, one has Mor(I/P) = Mor(I)\P, the
complement set of Pin Mor(I).
Let ιP:I/P֒→Ibe the inclusion functor. According to Lemma 5.1, the I-diagram Dof abelian
categories induces an I/P-diagram D◦ιPof abelian categories. We construct a lift functor lifPfrom
(D◦ιP)-Rep to D-Rep, which roughly speaking, is obtained by adding zeroes. Explicitly, given an
object Mand a morphism σ:M→M′in (D◦ιP)-Rep, define
•for i∈Ob(I),
lifP(M)i=(Mi,if i∈Ob(I/P);
0,otherwise;
•for α:i→jin Mor(I),
lifP(M)α=(Mα:Dα(Mi)→Mj,if α∈Mor(I/P),
0,otherwise;
•for i∈Ob(I),
lifP(σ)i=(σi,if i∈Ob(I/P),
0,otherwise.
The following proposition tells us that what we constructed above is indeed a functor.
6.1 Proposition. Let Pbe a prime ideal of I. Then lifP: (D◦ιP)-Rep →D-Rep is a functor.
Proof. We show that lifPsends objects to objects and morphisms to morphisms. Other axioms of
functors can be verified routinely. To show that lifP(M) is an object in D-Rep, we need to check
that lifP(M) satisfies the axioms (lRep.1) and (lRep.2) in Definition 2.1.
For the axiom (lRep.1), we want to prove the equality
lifP(M)βα ◦τβ,α (lifP(M)i) = lifP(M)β◦Dβ(lifP(M)α)
for any pair iα
→jβ
→kof composable morphisms in Mor(I). We have four cases.
(a) If α∈Mor(I/P) and β∈Mor(I/P), then α /∈Pand β /∈P, so βα ∈Mor(I/P) since Mor(I/P) =
Mor(I)\Pis closed under composition of morphisms. In this case the desired equality holds
as Msatisfies the axioms (lRep.1).
(b) If α∈Mor(I/P) but β /∈Mor(I/P), then β∈Pand lifP(M)β= 0, so lifP(M)β◦Dβ(lifP(M)α) =
0. On the other hand, since Pis an ideal, βα ∈P, so βα /∈Mor(I/P) and lifP(M)βα = 0.
Consequently, lifP(M)βα ◦τβ,α (lifP(M)i) = 0, as desired.
(c) If α /∈Mor(I/P) but β∈Mor(I/P), then one can verify the equality as in the second case.
(d) If α /∈Mor(I/P) and β /∈Mor(I/P), then lifP(M)α= 0 and lifP(M)β= 0, so lif P(M)β◦
Dβ(lifP(M)α) = 0. On the other hand, βα /∈Mor(I/P) because Pis closed under composition,
and hence lifP(M)βα = 0, which implies the desired equality.
10This is well defined since in this case eiand ejcan not be in Pas Pis a two-sided ideal, and hence iand jare
objects in Ob(I/P).
40 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
For the axiom (lRep.2), we want to prove the equality
lifP(M)ei◦ηi(lifP(M)i) = idlifP(M)i
for i∈Ob(I). Indeed, if i∈Ob(I/P), then ei∈Mor(I/P), so we have
lifP(M)ei◦ηi(lif P(M)i) = Mei◦ηi(Mi) = idMi= idlif P(M)i,
where the second equality holds as Msatisfies the axiom (lRep.2). In the case where i /∈Ob(I/P),
we have ei/∈Mor(I/P) and lif P(M)i= 0, so
lifP(M)ei◦ηi(lifP(M)i) = 0 = idlifP(M)i.
Thus, lifP(M)αsatisfies the axiom (lRep.2) as well.
To verify that lifP(σ) is a morphism in D-Rep, we have to show
lifP(σ)j◦lifP(M)α=lifP(M′)α◦Dα(lifP(σ)i)
for all α:i→j∈Mor(I). Indeed, if α∈Mor(I/P), then both iand jare in Ob(I/P), and we
deduce that
lifP(σ)j◦lifP(M)α=σj◦Mα=M′
α◦Dα(σi) = lifP(M′)α◦Dα(lifP(σ)i).
If α /∈Mor(I/P), then both lifP(M)αand lifP(M′)αare zero. The desired equality holds clearly in
this case.
6.2 Remark. The natural inclusion functor ιP:I/P֒→Iinduces a restriction functor
ι∗
P:D-Rep →(D◦ιP)-Rep
by Proposition 5.2. Clearly, ι∗
P◦lifPis isomorphic to the identity functor on (D◦ιP)-Rep.
For any object Min D-Rep, by the universal property of colimits, there exists a unique morphism
ϕM
isuch that for every morphism θ∈P(•, i), the diagram
(6.2.1) Dθ(Ms(θ))
sM
θ
Mθ
''
P
P
P
P
P
P
P
P
P
P
P
P
P
P
colimθ∈P(•,i)Dθ(Ms(θ))ϕM
i
//Mi
commutes, where sM
θis the canonical morphism, and the colimit is taken over the morphism
category whose object set is
P(•, i) = {θ∈Mor(I)|θ∈Pwith target i}
and morphisms between θ:• → iand θ′:∗ → iare morphisms δ:• → ∗ such that θ=θ′δ.
Suppose that Disatisfies the axiom AB3 for i∈Ob(I) and Dαpreserves small coproducts for
α∈Mor(I/P). We use the above commutative diagram to define the left adjoint cokP:D-Rep →
(D◦ιP)-Rep of the lift functor. Explicitly, given an object Mand a morphism ω={ωi}i∈Ob(I):
M→M′in D-Rep,
•for i∈Ob(I/P), set cokP(M)ito be coker(ϕM
i); see (6.2.1). Explicitly,
cokP(M)i=Mi/X
θ∈P(•,i)
Mθ(Dθ(Ms(θ))).
We shall mention that the sum appearing in the above equality is precisely the image of
ϕM
idefined in (6.2.1).
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 41
•For α:i→j∈Mor(I/P), one can obtain a composite of morphisms
MαDα(X
θ∈P(•,i)
Mθ(Dθ(Ms(θ))))→X
θ∈P(•,i)
Mαθ(Dαθ(Ms(θ))) →X
θ∈P(•,j)
Mθ(Dθ(Ms(θ)))
where the first one is obtained by applying a suitable natural transformation in (l.Rep1)
and the second one is an actual inclusion. This composite of morphisms gives rise to a
morphism αmaking the following diagram
(6.2.2) Dα(colimθ∈P(•,i)Dθ(Ms(θ)))
Dα(ϕM
i)
//Dα(Mi)
Dα(πM
i)
//
Mα
Dα(cokP(M)i)//
α
0
colimθ∈P(•,j)Dσ(Ms(σ))ϕM
j
//Mj
πM
j
//cokP(M)j//0
commutes. We then define cokP(M)αto be α.
•For i∈Ob(I/P), one has
πM′
i◦ωi◦ϕM
i◦sM
θ=πM′
i◦ωi◦Mθ=πM′
i◦M′
θ◦Dθ(ωs(θ)) = πM′
i◦ϕM′
i◦sM′
θ◦Dθ(ωs(θ)) = 0,
where the first and third equalities follow from (6.2.1). By the universal property of colimts,
πM′
i◦ωi◦ϕM
i= 0. Therefore, by the universal property of cokernels, we can find a unique
morphism cokP(ω)isuch that the diagram
(6.2.3) colimθ∈P(•,i)Dθ(Ms(θ))ϕM
i//Mi
πM
i//
ωi
cokP(M)i//
cokP(ω)i
0
colimθ∈P(•,i)Dθ(M′
s(θ))ϕM′
i//M′
i
πM′
i//cokP(M′)i//0
commutes.
6.3 Theorem. Let Pbe a prime ideal of I. Suppose that Disatisfies the axiom AB3 for i∈Ob(I/P)
and Dαpreserves small coproducts for α∈Mor(I/P). Then cokPdefined above is a functor, and is
the left adjoint of lifP.
Proof. To establish the first statement, it is enough to show that cokPsends objects to objects
and morphisms to morphisms, and other axioms of functors are clear since the above construction
of cokPis functorial. To show that cokP(M) is an object in (D◦ιP)-Rep, we need to verify the
axioms (lRep.1) and (lRep.2) in Definition 2.1.
For any pair iα
→jβ
→kof morphisms in Mor(I/P), we have equalities
cokP(M)β◦Dβ(cokP(M)α)◦Dβ(Dα(πM
i))
=cokP(M)β◦Dβ(πM
j)◦Dβ(Mα) by (6.2.2)
=πM
k◦Mβ◦Dβ(Mα) by (6.2.2)
=πM
k◦Mβα ◦τβ,α(Mi) by (lRep.1)
=cokP(M)βα ◦Dβα (πM
i)◦τβ,α(Mi) by (6.2.2)
=cokP(M)βα ◦τβ,α(cokP(M)i)◦Dβ(Dα(πM
i)),
where the last equality follows from the commutative diagram obtained by applying τβ,α to πM
i.
Since both Dβand Dαare right exact and πM
iis an epimorphism, so is Dβ(Dα(πM
i)). Thus we
have
cokP(M)β◦Dβ(cokP(M)α) = cokP(M)βα ◦τβ,α (cokP(M)i).
42 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
For i∈Ob(I/P), we have equalities
cokP(M)ei◦ηi(cokP(M)i)◦πM
i
=cokP(M)ei◦Dei(πM
i)◦ηi(Mei)
=πM
i◦Mei◦ηi(Mei) by (6.2.2)
=πM
i,
where the first equality holds by the commutative diagram obtained by applying ηito πM
i. Since
πM
iis an epimorphism, one has
cokP(M)ei◦ηi(cokP(M)i) = idcokP(M)i,
which is exactly the axiom (lRep.2).
To verify that cokP(ω) = {cokP(ω)i}i∈Ob(I/P)is a morphism from cokP(M) to cokP(M′), we must
show the equality
cokP(ω)j◦cokP(M)α=cokP(M′)α◦Dα(cokP(ω)i)
for any morphism α:i→j∈Mor(I/P). But we have
cokP(ω)j◦cokP(M)α◦Dα(πM
i)
=cokP(ω)j◦πM
j◦Mαby (6.2.2)
=πM′
j◦ωj◦Mαby (6.2.3)
=πM′
j◦M′
α◦Dα(ωi)
=cokP(M′)α◦Dα(πM′
i)◦Dα(ωi) by (6.2.2)
=cokP(M′)α◦Dα(cokP(ω)i)◦Dα(πM
i),by (6.2.3)
where the third equality holds as ωis a morphism in D-Rep. Since Dαis right exact and πM
iis an
epimorphism, it follows that Dα(πM
i) is also an epimorphism. Hence, the desired equality follows.
Now we prove the second statement. Let Mbe an object in D-Rep and Nan object in (D◦
ιP)-Rep. We construct a pair of natural maps
u: Hom(D◦ιP)-Rep(cokP(M), N )⇄HomD-Rep(M, lifP(N)) : v
which are inverse to each other. Note that for i∈Ob(I/P), there exists an exact sequence
colim
θ∈P(•,i)
Dθ(Ms(θ))ϕM
i
−→ Mi
πM
i
−→ cokP(M)i→0
in Di.
To define the map u, let σ={σi}i∈Ob(I/P):cokP(M)→Nbe a morphism in (D◦ιP)-Rep. Then
for i∈Ob(I), define
u(σ)i=(σi◦πM
i,if i∈Ob(I/P),
0,otherwise
by noting that lifP(N)i=Niif i∈Ob(I/P) and Ni= 0 otherwise.
To show that u(σ) = {u(σ)i}i∈Ob(I)is a morphism from Mto lifP(N), we verify the equality
u(σ)j◦Mα=lifP(N)α◦Dα(u(σ)i) (♯)
for any α:i→j∈Mor(I) case by case.
(a) If j /∈Ob(I/P), then lifP(N)j= 0, and the equality (♯) holds.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 43
(b) If j∈Ob(I/P) but i /∈Ob(I/P), then ei∈P, so α∈P(•, j ). Consequently, Mα=ϕM
i◦sM
α,
and one has
u(σ)j◦Mα=σj◦πM
j◦Mα=σj◦πM
j◦ϕM
i◦sM
α= 0.
On the other hand, since u(σ)i= 0 by definition, lifP(N)α◦Dα(u(σ)i) = 0. This implies that
the equality (♯) holds, too.
(c) If j∈Ob(I/P), i∈Ob(I/P) but α /∈Mor(I/P), one can check the equality by an argument
similar to that of the previous case.
(d) If j∈Ob(I/P), i∈Ob(I/P) and α∈Mor(I/P), then one has
lifP(N)α◦Dα(u(σ)i) = Nα◦Dα(σi)◦Dα(πM
i)
=σj◦cokP(M)α◦Dα(πM
i)
=σj◦πM
j◦Mα
=u(σ)j◦Mα,
where the first equality holds by the definitions of lifP(N)αand u(σ)i.
To define the map v, let ω={ωi}i∈Ob(I):M→lifP(N) be a morphism in D-Rep. Then for
i∈Ob(I/P) and θ∈P(•, i), one has
ωi◦ϕM
i◦sM
θ=ωi◦Mθ=lifP(N)θ◦Dθ(ωs(θ)) = 0,
where the first equality holds by (6.2.1), and the last equality holds since θ /∈Mor(I/P). Therefore,
ωi◦ϕM
i= 0 by the universal property of colimts, and hence we can find a unique morphism v(ω)i
such that the diagram
(6.3.1) colimθ∈P(•,i)Dθ(Ms(θ))ϕM
i//Mi
πM
i//
ωi
cokP(M)i
v(ω)i
ww♣♣♣♣♣♣♣♣♣♣♣
//0
lifP(N) = Ni
commutes.
We show that v(ω) = {v(ω)i}i∈Ob(I/P)is a morphism from cokP(M) to Nby checking the equality
v(ω)j◦cokP(M)α=Nα◦Dα(v(ω)i)
for any morphism α:i→j∈Mor(I/P). Indeed, one has
v(ω)j◦cokP(M)α◦Dα(πM
i) = v(ω)j◦πM
j◦Mαby (6.2.2)
=ωj◦Mαby (6.3.1)
=lifP(N)α◦Dα(ωi)
=Nα◦Dα(v(ω)i)◦Dα(πM
i) by (6.3.1).
Since Dα(πM
i) is an epimorphism, the desired equality follows.
Clearly, the maps vand uare natural with respect to Mand N, so it remains to prove that
they are inverse to each other. For this purpose, let ibe an object in Ob(I). If i /∈Ob(I/P), then
u(v(ω))i= 0 by definition. Note that lif P(N)i= 0 in this case, so ωi= 0. Therefore, u(v(ω))i=ωi.
If i∈Ob(I/P), then u(v(ω))i=v(ω)i◦πM
i=ωi, where the the second equality holds by (6.3.1).
Consequently, we always have uv(ω) = ω.
On the other hand, one has
v(u(σ))i◦πM
i=u(σ)i=σi◦πM
i,
44 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
where the first equality holds by (6.3.1). By the universal property of cokernels, we conclude that
v(u(σ))i=σi, and hence vu(σ) = σ.
6.2. The stalk functor and its left adjoint. In this subsection we apply the general results
established in the previous subsection to a special type of index categories I:partially ordered
categories. In this case, the subset Pi=Mor(I)\EndI(i) is a prime ideal for all i∈Ob(I), so I/Pi
coincides with the full subcategory of Iconsisting of the single object i. Consequently, under some
extra assumptions, the lift functor and cokernel functor constructed in the previous subsection give
other relations between D-Rep and Di, besides the relations provided by evaluation functor and its
left adjoint.
We recall the notion of partially ordered categories. The relation 4on the set Ob(I) such that
i4jif HomI(i, j)6=∅is clearly reflexive and transitive, but in general not anti-symmetric. If
furthermore 4is anti-symmetric, then Ob(I) becomes a poset with respect to the partial order
4. Categories satisfying this condition are called partially ordered categories (or weakly directed
categories,directed categories in the literature by some authors), which can be viewed as a natural
generalization of posets and quivers without oriented cycles (but loops from a vertex to itself are
allowed).
In this subsection we always suppose that Iis a partially ordered category. Then Mor(I)\EndI(i)
forms a prime ideal for any i∈Ob(I), which is denoted by Pifor short. It is also clear that I/Pi
is the full subcategory of Iwith only one object i. Therefore, we have canonical inclusion functors
ιi:Ii→I,i:Ii→I/Piand ιPi:I/Pi→I. Clearly, we have ιi=ιPi◦i.
Suppose that Disatisfies the axiom AB3 and Dγpreserves small coproducts for γ∈EndI(i).
Then we have the following functors:
D-Rep cokPi//(D◦ιPi)-Rep ∗
i//(D◦ιi)-Rep ∼
=Di,
whose composite is denoted by coki. Explicitly, given a representation Mover D, one has
coki(M) = Mi/X
θ∈Pi(•,i)
Mθ(Dθ(Ms(θ))).
The right adjoint of cokican be defined as follows. The right adjoint of cokPiis the functor
lifPi: (D◦ιPi)-Rep →D-Rep.
If Disatisfies the axiom AB3∗, we can construct the right adjoint
rani: (D◦ιi)-Rep →(D◦ιPi)-Rep,
of ∗
iwhich sends an object Kin Di∼
=(D◦ιi)-Rep to the object
rani(K) = Y
γ∈EndI(i)
Dγ(K)
in (D◦ιPi)-Rep. Thus we define the stalk functor at ias the composite of
stai:Di∼
=(D◦ιi)-Rep rani//(D◦ιPi)-Rep lifPi
//D-Rep.
Explicitly, given an object Mi∈Diand for j∈Ob(I), one has
(stai(Mi))j=(0,if i6=j;
Qθ:i→iDθ(Mi),else.
6.4 Remark. We have actually defined two cokernel functors cokiand cokPi. Given an object M
in D-Rep,cokPi(M) is an object in (D◦ιPi)-Rep, which can be also identified with an object in
Di. With this identification, one can see that coki(M) = cokPi(M). This is not surprising since
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 45
cokiis the composite of cokPiand the restriction functor, which does not change the value. On the
other hand, note that an object Min (D◦ιPi)-Rep uniquely determines an object Mi∈Di, so the
functor lifPican also be viewed as a stalk functor since one has
(lifPi(M))j=(0,if i6=j;
Mi,else,
which is different from stai(Mi). The reason is very subtle: when applying lifPiwe regard Mias
a representation over D◦ιPi, while applying staiwe only regard Mias an object in Diand hence
forget the actions of all non-identity morphisms θ:i→ion it.
The following result follows immediately from Theorem 6.3.
6.5 Corollary. Let ibe an object in Ob(I). Suppose that Iis a partially ordered category, Di
satisfies both the axioms AB3 and AB3∗, and Dγpreserves small coproducts for γ∈EndI(i). Then
(coki,stai)is an adjoint pair.
6.6 Example. The stalk functor and its left adjoint have been widely used in the literature, see
e.g. [18, 19, 23, 24, 40, 41, 64]. In particular, when Iis the free category associated to a quiver
Q= (Q0, Q1) without oriented cycles, these two functors have a very simple description. For
instance, when Dis a trivial Q-diagram over a fixed abelian category Aand V:Q→Ais a
functor, then coki(V) is the quotient Vi/Pθ∈Q1(•,i)θ(V•), where Q1(•, i) is the set of arrows with
target i.
6.7 Remark. If D◦ιPiis locally exact (i.e., Dγis an exact functor on Difor every γ∈EndI(i)),
and Disatisfies the axiom AB4∗, then raniis exact as well. Consequently, staiis exact since so is
lifPi.
6.3. Dual results for Rep-D.Throughout this subsection we suppose that Pis a prime ideal of
Mor(I) and Dis left exact, and briefly describe dual constructions and results for the category
Rep-D.
The natural inclusion functor ιP:I/P֒→Iinduces an I/P-diagram D◦ιPas well as a lift functor
lifP:Rep-(D◦ιP)→Rep-D.
Suppose that Disatisfies the axiom AB3∗. Then for any object Kin Rep-D, by the universal
property of limits, there exists a unique morphism ψK
isuch that the diagram
(6.7.1) Ki
Kθ''
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
ψK
i//limθ∈P(•,i)Dθ(Ks(θ))
πK
θ
Dθ(Ks(θ))
commutes, where πK
θdenotes the canonical morphism. We can use this commutative diagram to
define a functor kerPdual to the one described in Subsection 6.1, and obtain following result dual
to Theorem 6.3.
6.8 Theorem. Suppose that Disatisfies the axiom AB3∗for i∈Ob(I/P)and Dαpreserves small
products for α∈Mor(I/P). Then kerPdefined above is a functor, and is the right adjoint of lifP.
Suppose that Iis a partially ordered category and Disatisfies the axiom AB3. Then one can
define an induction functor
(i)!:Di≃
−→ Rep-(D◦ιi)−→ Rep-(D◦ιPi)
46 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
which sends object Kin Dito the object
(i)!(K) = a
γ∈EndI(i)
Dγ(Ki)
in Rep-(D◦ιPi). The stalk functor at iis defined to be the composite
stai:Di≃
−→ Rep-(D◦ιi)(i)!
−→ Rep-(D◦ιPi)lifPi
−→ Rep-D.
When Disatisfies the axiom AB3∗and Dγpreserves small products for γ∈EndI(i), it has a right
adjoint, which is the composite
keri:Rep-DkerPi
−→ Rep-(D◦ιPi)∗
i
−→ Rep-(D◦ιi)≃
−→ Di.
An immediate consequence of Theorem 6.8 is the following result.
6.9 Corollary. Let ibe an object in Ob(I). Suppose that Iis a partially ordered category, Di
satisfies both axioms AB3 and AB3∗, and Dγpreserves small products for γ∈EndI(i). Then
(stai,keri)is an adjoint pair.
7. Rooted categories and characterizations of projectives and injectives
In this section, we introduce a few categories Isatisfying certain combinatorial conditions, including
left rooted categories, locally trivial categories, and direct categories. We clarify relations among
these categories, and give characterizations of projective objects (resp., injective objects) in D-Rep
(resp., Rep-D).
7.1. Rooted categories. We begin this subsection by introducing the following transfinite se-
quence inspired by the work [23] of Enochs, Oyonarte and Torrecillas.
7.1. Suppose that Iis a partially ordered category, that is, the relation 4on Mor(I) defined by
setting i4jif HomI(i, j)6=∅is a partial order. We define a transfinite sequence {Vχ}χordinal of
subsets of Ob(I) as follows:
•For the first ordinal χ= 0, set V0=∅.
•For a successor ordinal χ+ 1, set
Vχ+1 =i∈Ob(I)iis not the target of any α∈Mor(I)
with source s(α)6=iand s(α)/∈ ∪µ6χVµ.
•For a limit ordinal χ, set Vχ=∪µ<χVµ.
7.2 Remark. We can give the slightly mysterious definition of transfinite sequences a more trans-
parent interpretation. That is, V0=∅,V1consists of objects in Ob(I) which is minimal with respect
to the partial order 4, and Vχ+1 is the union of Vχand the set of minimal elements in Ob(I)\Vχ.
By this observation, one deduces that there is a chain V1⊆V2⊆ · · · ⊆ Ob(I).
A nonempty subset Iof a poset (P, 6) is said to be an ideal if the following is true: for j∈I
and i∈P, if i6j, then i∈Ias well.
7.3 Lemma. Suppose that Iis a partially ordered category. Then each Vχin the above construc-
tion with χ>1is an ideal of the poset (Ob(I),4).
Proof. Since V1is the set of minimal objects with respect to the partial order 4, it is obviously
an ideal. Now suppose that χ > 1. Take an object j∈Vχand an objetct i∈Ob(I) with ij,
that is, HomI(i, j)6=∅and i6=j. We want to show i∈Vχas well.
If χis a successor ordinal, then Vχ=Vχ−1∪Uχ, where Uχis the set of minimal objects in
Ob(I)\Vχ−1. We have two cases:
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 47
(a) If j∈Uχ, then jis a minimal object in Ob(I)\Vχ−1. Since ij, we deduce that i /∈
Ob(I)\Vχ−1, so i∈Vχ−1by Remark 7.2.
(b) If j /∈Uχ, then j∈Vχ−1. Note that Vχ−1is an ideal by induction hypothesis. It follows that
i∈Vχ−1as well, so i∈Vχby Remark 7.2.
If χis a limit ordinal, then Vχ=∪µ<χVµ, and hence jlies in a certain Vµ. By the induction
hypothesis, each Vµis an ideal, so i∈Vµ⊆Vχ.
7.4 Remark. In the above proof we actually obtain a stronger conclusion. That is, if χis a
successor ordinal, ij, and j∈Vχ, then i∈Vχ−1.
Now we are ready to define left rooted categories.
7.5 Definition. A partially ordered category Iis said to be left rooted if there exists an ordinal ζ
such that Vζ=Ob(I).
Let Ibe the free category associated to a quiver Q= (Q0, Q1). Explicitly, objects and morphisms
in Iare vertices and oriented paths (including the trivial paths) in Q. A morphism αin Iis called
aloop if it is an arrow from a vertex to itself, and is an oriented cycle if it is a oriented path from
a vertex to itself but is not an arrow. Clearly, Iis a partially ordered category if and only if the
associated quiver Qhas no oriented cycles (but loops may exist in Qsince they contribute nothing
to the partial order 4).
7.6 Remark. We give a relation between rooted quiver introduced in [23] and rooted categories
introduced in this paper. Let Q= (Q0, Q1) be a quiver. There is a transfinite sequence {Vχ}χordinal
of subsets of Q0as follows:
•for the first ordinal χ= 0, set V0=∅;
•for a successor ordinal χ+ 1, set
Vχ+1 =i∈Q0iis not the target of any arrow α∈Q1
with s(α)/∈ ∪µ6χVµ;
•for a limit ordinal χ, set Vχ=∪µ<χVµ.
The quiver Qis called left rooted if there exists an ordinal ζsuch that Vζ=Q0.
Note that in the above definition of Vχ+1, we do not require s(α)6=i, which is imposed in the
definition of the transfinite sequences of partially ordered categories. This subtle difference causes
a big deviation. Indeed, if Qis left rooted, then it has no oriented cycles and loops (if α:i→i
is a loop, then the vertex iis not contained in any Vχ), and one can show that the associated free
category is a left rooted category. However, the converse statement is not true. That is, we can
find quivers which are not left rooted quivers, but their associated free categories are left rooted
categories. A simple example is the quiver with one vertex and one loop.
The underlying reason causing this difference is the definition of 4, which gives a possible partial
order on Q0. Indeed, we define i4jif there is an oriented path from ito j. As we mentioned
before, loops from a vertex ito itself do not bring any trouble to this possible partial order since we
always have i4i. Therefore, in our definition of left rooted categories, loops are allowed. However,
the definition of left rooted quivers in [23] implicitly implies that ijif there is an arrow from i
to j. Consequently, loops are not allowed since otherwise one will obtain ii, which is absurd.
The following result from [23, Proposition 3.6] gives a characterization of left rooted quivers.
7.7 Lemma. A quiver Qis left rooted if and only if it has no infinite sequence of arrows of the
form · · · → • → • → • (vertices in this sequence need not to be distinct).
7.8 Remark. Recall that a poset is said to be artinian if it satisfies the descending chain condition.
By a similar proof as in [23, Proposition 3.6], we conclude that a partially ordered category Iis left
rooted if and only if the poset (Ob(I),4) is artinian.
48 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Now we recall the definition of locally trivial categories studied in [58], which can be seen as a
generalization of posets.
7.9 Definition. A skeletal small category Iis called locally trivial if EndI(i) contains only the
identity morphism for all i∈Ob(I).
Direct categories play a prominent role in this paper for constructing cotorsion pairs. The
following definition is taken from [42, Definition 5.1.1].
7.10 Definition. A skeletal small category Iis called a direct category if there exists a functor
F:I→χ, where χis an ordinal (viewed as a category in a natural way) such that Fsends non-
identity morphisms in Ito non-identity morphisms in χ. Similarly, one can define direct quivers.
The following result clarifies relations among notions introduced above, and will be used fre-
quently in the rest of the paper.
7.11 Proposition. Let Ibe a skeletal small category, and Q= (Q0, Q1)a quiver. Then one has:
(a) If Iis locally trivial, then it is partially ordered.
(b) Iis a direct category if and only if it is a locally trivial left rooted category.
(c) Qis a direct quiver if and only if it is a left rooted quiver, and if and only if the associated
free category is a direct category.
Proof. (a) If there exist objects iand jsuch that i4jand j4i, then one can find two morphisms
α:i→jand β:j→i. But this implies that both α◦βand β◦αare the identity morphisms.
Consequently, i∼
=j. But Iis skeletal, so i=j.
(b) Suppose that Iis a direct category and let F:I→χbe the functor in Definition 7.10. If
there exists an object x∈Ob(I) and a non-identity morphism f:x→x, then F(x)< F (x) by the
definition of direct categories, which is absurd. Thus for every object x, there exists no non-identity
morphisms from xto itself, that is, Iis locally trivial. Furthermore, by defining
Vλ={x∈Ob(I)|F(x)6λ},
one can construct the desired transfinite sequence described at the beginning of this subsection.
Clearly, one has Vχ=Ob(I), so Iis left rooted.
Conversely, suppose that Iis a locally trivial left rooted category. Then there exists an ordinal
χand a transfinite sequence {Vλ}λ6χsuch that Vχ=Ob(C). Now we define another sequence of
subsets of Ob(C) as follows: U0=∅, and
Uλ=Vλ\([
µ<λ
Vµ).
Equivalently, Uλ+1 =Vλ+1 \Vλif λ+ 1 is a successor order, and Uλ=∅if λis a limit order. Then
one has
Ob(C) = G
λ6χ
Uλ.
Note that objects in each Uλare disjoint, that is, there is no morphisms among these objects. Now
define a functor F:I→χsuch that F(x) = λxfor x∈Uλxand Fsends a morphism x→yto
the unique morphism λx6λy. It is not hard to check that Fis well defined. Furthermore, since
Iis locally trivial, Fmaps non-identity morphisms to non-identity morphisms. Thus Iis a direct
category.
(c) Applying the argument in part (b) we can show that Qis a direct quiver if and only if it
is a left rooted quiver. For the second equivalence, let Ibe the free category associated to Q. By
Remark 7.6, if Qis a left rooted quiver, then Iis a left rooted category. The converse holds if I
is locally trivial. That is, Qis a left rooted quiver if and only if Iis a locally trivial left rooted
category.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 49
We end this subsection by giving a few examples.
7.12 Example. Let N∞be the following poset: elements are sequences n= (n1, n2,...) of natural
numbers such that only finitely many entries are nonzero, and m4nif and only if mi6ni
for every i. Then N∞is a direct category. Note that it is not a free category associated to a
quiver. Moreover, one can obtain an explicit isomorphism between this poset and the poset of
positive integers and division, by ordering all prime numbers in the natural order and applying the
fundamental theorem of arithmetics.
7.13 Example. Any skeletal full subcategory of the category of finite sets and injections is a left
rooted category. Similarly, let Abe an associative ring. Then any skeletal full subcategory of the
category of free left A-modules with finite rank and A-linear injections is a left rooted category. A
skeleton of the category of finitely generated left A-modules and injective module homomorphisms
is a left rooted category if and only if Ais left artinian. In general these categories are not locally
trivial.
7.2. Characterizations of projectives and injectives. Throughout this subsection we suppose
that Iis a partially ordered category. We apply the functors constructed in the previous sections
to characterize projective objects in D-Rep and injective objects in Rep-D.
We mention that for each i∈Ob(I), Pi=Mor(I)\EndI(i) is a prime ideal of Mor(I). We have
an inclusion functor ιPi:I/Pi→I, which induces a restriction functor ι∗
Pi:D-Rep →(D◦ιPi)-Rep
and an induction functor (ιPi)!: (D◦ιPi)-Rep →D-Rep. The two cokernel functors cokiand cokPi
are defined in Section 6.
The following lemma is straightforward.
7.14 Lemma. Fix i∈Ob(I). Suppose that Disatisfies the axiom AB3 and Dγpreserves small
coproducts for γ∈EndI(i). If Pis a projective object in D-Rep, then cokPiis a projective object in
(D◦ιPi)-Rep. If furthermore Disatisfies the axiom AB4∗and D◦ιPiis locally exact, then coki(P)
is a projective object in Di.
Proof. The conclusion follows from the fact that (coki,stai) (resp., (cokPi,lif Pi)) is an adjoint pair
and stai(resp., lifPi) is exact; see Remark 6.7.
Given a family X={Xi}i∈Ob(I)with each Xia subcategory of Di, we use the functors cok•to
define a subcategory Φ(X) of D-Rep. This subcategory will play a key role for us to construct
cotorsion pairs in D-Rep and to characterize projective, flat and Gorenstein flat objects.
7.15 Definition. Suppose that Disatisfies the axiom AB3 for i∈Ob(I), and Dγpreserves small
coproducts for γ∈EndI(i). Define a subcategory of D-Rep:
Φ(X) = {X∈D-Rep |ϕX
iis a monomorphism and coki(X)∈Xifor i∈Ob(I)},
where ϕX
i: colimθ∈Pi(•,i)Dθ(Xs(θ))→Xiis given in (6.2.1)11. In particular,
Φ(D) = {X∈D-Rep |ϕX
iis a monomorphism for i∈Ob(I)}.
The next result can be checked routinely.
7.16 Lemma. Suppose that Disatisfies the axioms AB3 for i∈Ob(I), and Dγpreserves small
coproducts for γ∈EndI(i). Then:
(a) The subcategory Φ(D)is closed under direct summands.
(b) If further Disatisfies the axiom AB4 for i∈Ob(I)and Dαpreserves small coproducts for
α∈Mor(I), then Φ(D)is closed under small coproducts.
11Under the assumption that Iis partially ordered, Piis a prime ideal for i∈Ob(I). Therefore, if each Disatisfies
the axiom AB3, then one can define ϕX
i.
50 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
The following result gives a characterization of projective objects in D-Rep.
7.17 Theorem. Suppose that Disatisfies the axiom AB4 for i∈Ob(I)and Dαpreserves small
coproducts for α∈Mor(I). Let Hbe an object in D-Rep. If His projective, then the following
conditions hold for every j∈Ob(I):
(a) ϕH
j: colimσ∈Pj(•,j)Dσ(Hs(σ))→Hjis a monomorphism and
(b) cokPj(H)is a projective object in (D◦ιPj)-Rep.
If furthermore Iis left rooted, then the converse holds.
Proof. The statement (b) follows from Lemma 7.14. In the following we prove the statement (a),
that is, H∈Φ(D). We mention that any projective object in D-Rep is isomorphic to a direct
summand of a direct sum of members in the family {frei(Pi)|i∈Ob(I) and Pi∈Proj(Di)}; see
Remark 5.10. Thus by Lemma 7.16, it suffices to prove that frei(Mi) is in Φ(D) for any object Mi
in Di.
Denote frei(Mi) by Lfor short. It is clear that ϕL
iis a monomorpism. In the following we prove
that the morphism
ϕL
j: colim
σ:t→j
t6=j
Dσ(Lt) = colim
σ:t→j
t6=j
Dσ(a
θ:i→t
Dθ(Mi)) −→ a
α:i→j
Dα(Mi) = Lj
is actually an isomorphism for each j∈Ob(I) with j6=i, where we write
colim
σ∈Pj(•,j)= colim
σ:t→j
t6=j
which looks more intuitive. By the universal property of colimits, we only need to show that Ljis
isomorphic to the colimit on the left side.
Recall that a morphism δ:t→t′,t6=j6=j′in the morphism category with object set P(•, j) is
a commutative diagram
t
σ
❀
❀
❀
❀
❀
❀
❀
❀
δ//t′
σ′
✁
✁
✁
✁
✁
✁
✁
✁
j
In particular, δinduces an obvious morphism
`θ:i→tDσ′(Dδθ(Mi)) δ∗
//`θ′:i→t′Dσ′(Dθ′(Mi))
by sending each component in the coproduct of the left side to the same component in the coproduct
of the right side since δθ is a morphism from i→t′, which is clearly a monomorphism. We also
note that
Dσa
θ:i→t
Dθ(Mi)∼
=a
θ:i→t
Dσ(Dθ(Mi)) ∼
=a
θ:i→t
Dσ′(Dδθ (Mi))
and a
θ′:i→t′
Dσ′(Dθ′(Mi)) ∼
=Dσ′a
θ′:i→t′
Dθ′(Mi)
since Dαpreserves small colimits for each α∈Mor(I). Consequently, we obtain a morphism
Dσ`θ:i→tDθ(Mi)δσ,σ′
//Dσ′`θ′:i→t′Dθ′(Mi).
On the other hand, for each σ:t→jand any θ:i→tin Mor(I), one has a composite of morphisms
ψσ:Dσa
θ:i→t
Dθ(Mi)∼
=a
θ:i→t
Dσ(Dθ(Mi)) −→ a
α:i→j
Dα(Mi)
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 51
where the second morphism sends each Dσ(Dθ(Mi)) to Dσθ(Mi). This in general is not monic.
Now we assemble these maps to obtain the following diagram
Dσ`θ:i→tDθ(Mi)δσ,σ′
//
ψσ))
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
Dσ′`θ′:i→t′Dθ′(Mi)
ψσ′
uu❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥
`α:i→jDα(Mi),
which is clearly a cocone indexed by the over category ιPj/j. It is also easy to see that the colimit
of this cocone is exactly colimσ:t→j
t6=j
Dσ(Lt), so it suffices to show that the above cocone is a limit
cocone to deduce the desired isomorphism between colimσ:t→j
t6=j
Dσ(Lt) and `α:i→jDα(Mi).
Take an arbitrary cocone indexed by ιPj/j as follows:
Dσ`θ:i→tDθ(Mi)δσ,σ′
//
φσ
''
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
Dσ′`θ′:i→t′Dθ′(Mi)
φσ′
vv♥♥♥♥♥♥♥♥♥♥♥♥♥♥
X,
For each α:i→j, one can define a map ραwhich is the composite:
Dα(Mi)−→ a
ς:i→i
Dα(Dς(Mi)) ∼
=Dαa
ς:i→i
Dς(Mi)−→ X
where the first map is the inclusion (by taking ςbe the identity on i) and the last one is φα. By
the universal property of coproducts, one gets a unique morphism
ρ:a
α:i→j
Dα(Mi)−→ X
such that ρrestricted to each component is exactly ρα.
We need to verify that the following diagram
Dσ`θ:i→tDθ(Mi)ψσ//
φσ
))
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚`α:i→jDα(Mi)
ρ
X
commutes for each σ. It suffices to show the commutativity of the following diagram for each θ:
(7.17.1) Dσ(Dθ(Mi)) ǫ//`θ:i→tDσ(Dθ(Mi)) ∼
=Dσ`θ:i→tDθ(Mi)ψσ//
φσ
++
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲
❲`α:i→jDα(Mi)
ρ
X
where ǫis the natural inclusion. By our constructions, ρψσǫequals to the composite of the following
maps
Dσ(Dθ(Mi)) ∼
=//Dσθ (Mi)inc //`ς:i→iDσθ (Dς(Mi)) ∼
=Dσθ `ς:i→iDς(Mi)φσθ //X
52 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
where the map inc is obtained by taking ςto be the identity on i. By the commutativity of the
cocone, φσθ =φσδσθ,σ, so ρψσǫequals to the composite
Dσ(Dθ(Mi)) ∼
=Dσθ (Mi)inc //`ς:i→iDσθ (Dς(Mi)) ∼
=Dσθ `ς:i→iDς(Mi)δσθ,σ
//Dσ`θ:i→tDθ(Mi)
φσ
X.
This establishes the commutativity of the diagram (7.17.1). Furthermore, it is also clear that the
map ρguaranteeing the commutativity of (7.17.1) must be defined as above. Thus as claimed,
colim
σ:t→j
t6=j
Dσ(Lt)∼
=a
α:i→j
Dα(Mi).
Now we suppose that Iis a left rooted category. Let Hbe an object in D-Rep satisfying
statements (a) and (b), and let {Vχ}be the transfinite sequence of subsets of Ob(I) defined in 7.1.
Then Ob(I) = Vζfor a certain ordinal ζ. We construct a family {Hχ}χ6ζof projective subobjects
of Has follows.
Set H0= 0 and H1=`i∈V1(ιPi)!(Hi). Note that objects in V1are minimal. Consequently, for
i∈V1, one has Hi=cokPi(H) which is projective in (D◦ιPi)-Rep. Since (ιPi)!is the left adjoint of
the exact functor ι∗
Pi, it preserves projective objects. Therefore, (ιPi)!(Hi) is projective in D-Rep,
and hence H1is also projective in D-Rep.
There are two cases:
(a) Suppose that χ+ 1 is a successor ordinal and let ibe an object in Vχ+1 \Vχ. Then we set
Hχ+1 =a
i∈Vχ+1\Vχ
(ιPi)!(cokPi(H)),
which is clearly is projective in D-Rep by the statement (b).
(b) Suppose that µis a limit ordinal and the projective objects Hκhave been constructed for all
ordinals κ < µ. Then set Hµ=`κ<µ Hκ, which is projective.
Now define Hζ=`χ<ζ Hχ. It is routine to check that His indeed isomorphic to Hζ, using the
following isomorphism
Hi∼
=( colim
θ∈Pi(•,i)
Dθ(Hs(θ))) ⊕cokPi(H).
Thus His a projective object in R-Rep, as desired.
7.18 Remark. The above proof of the first direction is constructive and very detailed. Actually,
one can give a much shorter proof from the abstract viewpoint. Let jbe an object in Isuch that
j6=iand there exists at least one morphism from ito j. Since Iis a partially ordered category, one
has ij. Denote by Ijthe full subcategory of Iconsisting of objects tsuch that tj. Clearly,
the object iis contained in Ij.
We have the following inclusion functors F:Ii→Ijand G:Ij→I, whose composite is
precisely ιi:Ii→I. Consequently, one has frei∼
=G!F!. We also note that for an object Kin (D◦
G)-Rep, one has (G!(K))j= colimG/j Dσ(Ks(σ)) = colimσ∈P(•,j)Dσ(Ks(σ)) = colimσ:t→j
t6=j
Dσ(Kt).
Therefore, for an object Miin Di,
a
α:i→j
Dα(Mi) = (frei(Mi))j∼
=(G!(F!(Mi)))j= colim
σ:t→j
t6=j
Dσ(F!(Mi))t.
But one also has (F!(Mi))t=`θ:i→tDθ(Mi) since F!is induced by the inclusion Ii→Ij. Conse-
quently, one has `α:i→jDα(Mi)∼
=colimσ:t→j
t6=j
Dσ`θ:i→tDθ(Mi)as claimed.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 53
Denote by Proj•the family {Proj(Di)}i∈Ob(I)with Proj(Di) the subcategory of Diconsisting of
projective objects in Di. Then one has:
7.19 Corollary. Suppose that Dis locally exact, Disatisfies both the axioms AB4 and AB4∗for
i∈Ob(I), and Dαpreserves small coproducts for α∈Mor(I). Then
Proj(D-Rep)⊆Φ(Proj•).
If furthermore Iis a direct category, then these two categories coincide.
Proof. Note that D◦ιPiis locally exact for every i∈Ob(I) as Dis locally exact by assumption.
By replacing cokPiwith coki, a similar version of statement (b) in the previous theorem holds by
Lemma 7.14. Now the same proof in the previous theorem shows that every projective object in
D-Rep also satisfies statement (a). This establishes the first conclusion. The second one is trivial
since in this case I/Pi=Ii, so (D◦ιPi)-Rep ≃Di,coki=cokPi, and frei= (ιPi)!.
7.20 Example. In Theorem 7.17 one cannot replace the functor cokPiby the functor coki. Here
we give a trivial example. Let Ibe a cyclic group of order 2 which is viewed as a category with
one object xand two morphisms, and let Dbe the trivial diagram with Dxthe category of vector
spaces over a field kof characteristic 2. Clearly, an object in D-Rep is nothing but a representation
of I. Let kbe the trivial representation. Then it clearly lies in Φ(Proj•), but is not a projective
representation of I. The reason is obvious: although an object Miin (D◦ιPi)-Rep can be viewed
as an object in Di, the condition that Miis projective in Dicannot guarantee that Miis also
projective in (D◦ιPi)-Rep.
We finish this section by describing some dual results for right representations.
7.21 Definition. Let Y={Yi}i∈Ob(I)with each member a subcategory of Di. Suppose that I
is a partially ordered category, Disatisfies the axiom AB3∗, and Dγpreserves small products for
γ∈EndI(i). Define
Ψ(Y) = {Y∈Rep-D|ψY
iis an epimorphism and keri(Y)∈Yifor i∈Ob(I)},
which makes sense since each Disatisfies AB3∗, so one can define ψY
ias in (6.7.1).
Denote by Inj•the family {Inj(Di)}i∈Ob(I)with Inj(Di) the subcategory of Diconsisting of injective
objects in Di. The following dual result of Theorem 7.17 gives a characterization of injectives in
Rep-D.
7.22 Theorem. Suppose that Disatisfies the axiom AB4∗for i∈Ob(I)and Dαpreserves small
products for α∈Mor(I). Let Ibe an object in Rep-D. If Iis injective, then the following conditions
hold for every j∈Ob(I):
(a) ψI
i:Ii→limθ∈Pi(•,i)Dσ(Is(θ))is an epimorphism and
(b) kerPi(I)is an injective object in Rep-(Di◦ιPi).
If furthermore Iis left rooted, then the converse holds.
Dual to Corollary 7.19, we have:
7.23 Corollary. Suppose that Dis locally exact, Disatisfies both the axioms AB4 and AB4∗for
i∈Ob(I), and Dαpreserves small products for α∈Mor(I). Then
Inj(D-Rep)⊆Ψ(Inj•).
If furthermore Iis a direct category, then these two categories coincide.
7.24 Remark. For the trivial diagram indexed by the free category associated to a left rooted
quiver, Corollaries 7.19 and 7.23 were proved by Enochs, Estrada and Garc´ıa Rozas in [18, Theorem
3.1] and [19, Theorem 4.2], respectively.
54 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
7.25 Remark. When Iis a direct category, according to the proofs of Theorem 7.17 and Corollary
7.19, for every projective object Min D-Rep, one has M∼
=`i∈Ob(I)frei(coki(M)). Dually, for every
injective object Nin Rep-D, one has N∼
=Qi∈Ob(I)frei(keri(N)). We mention that the two frei
appearing here have different meanings. Explicitly, the first freiis a functor from Dito D-Rep,
while the second freiis a functor from Dito Rep-D.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 55
Part III. Abelian model structures and homological objects
The main goal of this part is to describe a few applications of the functors introduced in Part
II, focusing on constructions of cotorsion pairs and abelian model structures on D-Rep as well as
characterizations of special homological objects in this category. We first recall some background
knowledge on cotorsion theory and abelian model category theory. Then we construct several co-
torsion pairs in D-Rep amalgamated from cotorsion pairs in those Di’s. In particular, we show
under some assumptions that the induced cotorsion pairs are complete, and that a family of hered-
itary abelian model structures on abelian categories Diamalgamates to a hereditary abelian model
structure on D-Rep. As special examples, we consider the categories R-Rep and Rep-R, where R
and Rare I-diagrams of module categories induced by an I-diagram Rof associative rings. Specif-
ically, we characterize Gorenstein homological objects in the two module categories, and construct
a Gorenstein injective model structure on Rep-Rand a Gorenstein flat model structure on R-Rep.
Setup. Throughout this part, unless otherwise specified, suppose that Dis a right exact I-diagram
such that Diis a Grothendieck category admitting enough projectives for each i∈Ob(I) and Dα
preserves small coproducts for each α∈Mor(I). Under this assumption, D-Rep is a Grothendieck
category admitting enough projectives by Theorem 5.11.
8. Preliminaries on cotorsion pairs and abelian model structures
In this section we give some background knowledge on cotorsion theory and abelian model category
theory. Throughout this section, let Abe an abelian category.
We begin with the following notions: a continuous direct (inverse) ζ-sequence in Aindexed by
an ordinal ζ, and an F-filtration (cofiltration) of an object in Awith respect to a subcategory F.
They will be used frequently in the later sections.
8.1 Definition. Let ζbe an ordinal. A continuous direct ζ-sequence in Ais a directed ζ-sequence
{Mχ, f χ′,χ}χ6χ′6ζof objects and morphisms in Asuch that fχ+1,χ is a monomorphism for any
ordinal χ < ζ and Mµ= colimχ<µ Mχfor any limit ordinal µ6ζ. Dually, a continuous inverse
ζ-sequence in Ais an inverse ζ-sequence {Mχ, gχ,χ′}χ6χ′6ζof objects and morphisms in Asuch
that gχ,χ+1 is an epimorphism for any ordinal χ < ζ and Mµ= limχ<µ Mχfor any limit ordinal
µ6ζ. Let Fbe a subcategory of A. An object Min Ais said to have an F-filtration if there is a
continuous direct ζ-sequence {Mχ, fχ′,χ}χ6χ′6ζsuch that M0= 0, Mζ=Mand coker(fχ+1,χ)∈F
for any ordinal χ < ζ. Dually, Mis said to have an F-cofiltration if there is a continuous inverse
ζ-sequence {Mχ, gχ,χ′}χ6χ′6ζsuch that M0= 0, Mζ=Mand ker(gχ,χ+1)∈Ffor any ordinal
χ < ζ.
For a subcategory Fof A, set
F⊥={M∈A|Ext1
A(F, M ) = 0 for F∈F}and ⊥F={M∈A|Ext1
A(M, F ) = 0 for F∈F}.
The following Lemma, known as Eklof’s Lemma, will be used in the proof of Lemma 9.8; see [17,
Theorem 1.2] and [61, Lemma 2.3] or [40, Lemmas 6.6 and 6.8].
8.2 Lemma. Let Fbe a subcategory of Aand Man object in A.
(a) If Asatisfies the axiom AB3 and Mhas an ⊥F-filtration, then Mis in ⊥F.
(b) If Asatisfies the axiom AB3∗and Mhas an F⊥-cofiltration, then Mis in F⊥.
Next, we recall the definition of (complete, hereditary) cotorsion pairs.
8.3 Definition. Let Fbe a subcategory of Aand Man object in A. A morphism f:F→M
is called a special F-precover of Mif fis an epimorphism, F∈Fand ker(f)∈F⊥. Dually, a
morphism g:M→Fis called a special F-preenvelope of Mif gis a monomorphism, F∈Fand
56 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
coker(g)∈⊥F. The subcategory Fis called special precovering (resp., special preenveloping) if
every object in Ahas a special F-precover (resp., a special F-preenvelope).
8.4 Definition. A pair (C,D) of subcategories of Ais called a cotorsion pair if C⊥=Dand
⊥D=C.
Let (C,D) be a cotorsion pair in A. Then Ais said to have enough C-objects (resp., D-objects)
if for any object Min A, there exists an epimorphism C։M(resp., a monomorphism MD)
with C∈C(resp., D∈D). Following from [21], a cotorsion pair (C,D) is said to have enough
projectives (resp., injectives) if Cis special precovering (resp., Dis special preenveloping) in A. A
cotorsion pair is called complete if it has both enough projectives and enough injectives.
A cotorsion pair (C,D) in Ais called resolving if Cis closed under taking kernels of epimorphisms
between objects of C, that is, for any short exact sequence 0 →X→Y→Z→0 in A, if Yand Z
are in C, then Xis in C. We say that (C,D) is coresolving if Dsatisfies the dual statement. Finally,
we say that a cotorsion pair is hereditary if it is both resolving and coresolving.
8.5 Remark. Let Cbe a subcategory of A. According to [33, Definition 2.2.1], there are two
associated cotorsion pairs (⊥(C⊥),C⊥) and (⊥C,(⊥C)⊥) in A, called the cotorsion pair generated by
Cand cogenerated by C, respectively.
The following result is a general form of the Salce’s Lemma; see [57] or [53, Lemma 3.3].
8.6 Lemma. Let (C,D)be a cotorsion pair in A. If one of the following conditions holds:
(a) (C,D)has enough projectives and Ahas enough D-objects (e.g., Ahas enough injectives);
(b) (C,D)has enough injectives and Ahas enough C-objects (e.g., Ahas enough projectives),
then (C,D)is complete.
Next, we recall the notion of abelian model structures and some relevant facts. A model category
is said to be abelian if its underlying category is abelian and the model structure is compatible with
the abelian structure of the underlying category (see Hovey [43] for details). The central result
on abelian model categories is now known as Hovey’s correspondence: a bijective correspondence
between complete and compatible cotorsion pairs and abelian model structures.
8.7 Theorem (Hovey’s correspondence). Let Abe a bicomplete abelian category. An abelian
model structure on Acorresponds bijectively to a triple (Q,W,R)of subcategories of Asuch that
(a) (Q,W∩R)and (Q∩W,R)are complete cotorsion pairs, and
(b) Wis thick, that is, it is closed under direct summands, extensions, and taking kernels of
epimorphisms and cokernels of monomorphisms.
Hovey’s correspondence makes it clear that an abelian model structure on Acan be succinctly
represented by the triple (Q,W,R) of subcategories of Athat satisfies the conditions (a) and (b)
in Theorem 8.7. Here, the objects in Q(resp., W,R) are actually cofibrant (resp., trivial,fibrant)
objects. Therefore, one often refers to such a triple as an abelian model structure in the literature,
and call it a Hovey triple.
Recall that a Hovey triple (Q,W,R) on Ais said to be hereditary if the complete cotorsion pairs
(Q,W∩R) and (Q∩W,R) are hereditary. The next result supplements Hovey’s correspondence,
making it much easier to construct abelian model structures from complete and hereditary cotorsion
pairs; see Gillespie [30, Theorem 1.1].
8.8 Lemma. Suppose that (Q,e
R)and (e
Q,R)are two complete and hereditary cotorsion pairs in
Asuch that e
Q⊆Q(or equivalently, e
R⊆R)and Q∩e
R=e
Q∩R. Then there exists a unique thick
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 57
subcategory Wsuch that (Q,W,R)is a Hovey triple. Moreover Wcan be characterized as
W={M|there is a s.e.s. 0→M→A→B→0with A∈e
Rand B∈e
Q}
={M|there is a s.e.s. 0→A′→B′→M→0with A′∈e
Rand B′∈e
Q}.
The reader can refer to [32] for more background materials on abelian model structures.
9. Induced cotorsion pairs in D-Rep
In this section we describe a method to amalgamate cotorsion pairs in those Dito obtain cotorsion
pairs in D-Rep. The principle technical tools for this amalgamation are the functors and adjunctions
constructed in Part II. We mention that Holm and Jørgensen established in [40] some cotorsion
pairs in D-Rep in the case where Dis a trivial diagram, that is, all Di’s are identical and each Dα
is the identity functor.
Setup. Throughout this section, let X={Xi⊆Di}i∈Ob(I)and Y={Yi⊆Di}i∈Ob(I)be two families
of subcategories such that (Xi,Yi) is a cotorsion pair in Difor i∈Ob(I).
We begin with the following result, which is from [40, Lemma 5.1].
9.1 Lemma. Let F:B⇄C:Gbe an adjoint pair between abelian categories Band C, and n>0
an integer. Suppose that the following statements hold for an object Xin Band an object Yin C:
(a) the functor Fsends any short exact sequence 0→G(Y)→X1→X→0in Bto a short exact
sequence 0→F(G(Y)) →F(X1)→F(X)→0in C;
(b) the functor Gsends any short exact sequence 0→Y→Y1→F(X)→0in Cto a short exact
sequence 0→G(Y)→G(Y1)→G(F(X)) →0in B.
Then there exists a natural isomorphism Ext1
C(F(X), Y )∼
=Ext1
B(X, G(Y)) of abelian groups.
The following result is an application of Lemma 9.1.
9.2 Lemma. Suppose that Dis exact. Then the following statements hold for each i∈Ob(I).
(a) The functor freiis exact;
(b) For every object Miin Diand every object M′in D-Rep, there exists a natural isomorphism
Ext1
D-Rep(frei(Mi), M ′)∼
=Ext1
Di(Mi,evai(M′)).
Proof. Statement (a) implies statement (b) by Lemma 9.1 since evaiis exact by Theorem 2.6, so
it suffices to prove statement (a).
Let Gi→Ki→Libe an exact sequence in Di. Since Dis exact by assumption, Dθis an exact
functor for any θ∈HomI(i, j). So we obtain an exact sequence
Dθ(Gi)→Dθ(Ki)→Dθ(Li)
in Dj. As Djsatisfies the axiom AB4, we get another exact sequence
a
θ∈HomI(i,j)
Dθ(Gi)→a
θ∈HomI(i,j)
Dθ(Ki)→a
θ∈HomI(i,j)
Dθ(Li)
in Dj, which is exactly
frei(Gi)j→frei(Ki)j→frei(Li)j.
By Theorem 2.6, freiis exact, as desired.
We introduce an extra notation. Given a family S={Si}i∈Ob(I)with each Sia subcategory of
Di, define a full subcategory D-RepSof D-Rep as follows:
D-RepS={M∈D-Rep |Mi∈Sifor all i∈Ob(I)}.
58 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Under some assumptions the subcategory D-RepYand its left-orthogonal subcategory in D-Rep
form a cotorsion pair in D-Rep.
9.3 Proposition. Suppose that Dis exact. Then (⊥D-RepY,D-RepY)is a cotorsion pair.
Proof. Let fre•(X) be the full subcategory of D-Rep whose objects are of the form frei(Xi) with
i∈Ob(I) and Xi∈Xi. By Remark 8.5, it suffices to show that D-RepY=fre•(X)⊥.
Let M′be an object in fre•(X) and Man object in D-RepY. Then there exists an object i∈Ob(I)
and an object Xi∈Xisuch that M′∼
=frei(Xi) and Mi∈Yi. By Lemma 9.2, we have
Ext1
D-Rep(M′, M )∼
=Ext1
D-Rep(frei(Xi), M )∼
=Ext1
Di(Xi,evai(M)) ∼
=Ext1
Di(Xi, Mi) = 0,
so D-RepY⊆fre•(X)⊥.
For the inclusion of the other direction, we take an object M′in fre•(X)⊥and show M′
i∈Yifor
i∈Ob(I). Let Xibe an object in Xi. Then by Lemma 9.2 again, we have
Ext1
Di(Xi, M ′
i)∼
=Ext1
Di(Xi,evai(M′)) ∼
=Ext1
D-Rep(frei(Xi), M ′) = 0.
It follows that M′
i∈X⊥
i=Yias (Xi,Yi) is a cotorsion pair in Di.
In the rest of this section we construct other cotorsion pairs in D-Rep if Iis partially ordered. It
follows from Corollary 6.5 that (coki,stai) is an adjoint pair for i∈Ob(I). This implies that cokiis
right exact for i∈Ob(I), so its first left derived functor L1cokiexists. If further Dis locally exact,
then staiis exact; see Remark 6.7. Thus the next result is a special case of [41, Lemma 1.2], which
is based on [56, Theorem 10.33].
9.4 Lemma. Suppose that Iis a partially ordered category and Dis locally exact. Then for
i∈Ob(I), object Yiin Diand object Xin D-Rep, there exists an exact sequence
0→Ext1
Di(coki(X), Yi)→Ext1
D-Rep(X, stai(Yi)) →HomDi(L1coki(X), Yi)→
Ext2
Di(coki(X), Yi)→Ext2
D-Rep(X, stai(Yi)).
Let S={Si}i∈Ob(I)be a family of subcategories of Di. Following [40, Definition 1.5], we define
a full subcategory Φ(S) of D-Rep as follows:
Φ(S) = {S∈D-Rep |L1coki(S) = 0 and coki(S)∈Sifor i∈Ob(I)}.
Our next result follows from [40, Theorem 1.7]. For the reader’s convenience, we give a proof.
9.5 Proposition. Suppose that Iis a partially ordered category and Dis locally exact. Then
(Φ(X),Φ(X)⊥)is a cotorsion pair in D-Rep.
Proof. Let sta•(Y) be the full subcategory of D-Rep whose objects are of the form stai(Yi) with
i∈Ob(I) and Yi∈Yi. We show that Φ(X) = ⊥sta•(Y). If this is true, then the conclusion follows
from Remark 8.5.
Let Xbe an object in Φ(X). Then coki(X)∈Xiand L1coki(X) = 0 for each i∈Ob(I), and so
for each object stai(Yi)∈sta•(Y), one has Ext1
D-Rep(X, stai(Yi)) = 0 by Lemma 9.4 as (Xi,Yi) is a
cotorsion pair in Di. Consequently, Xis an object in ⊥sta•(Y), so Φ(X)⊆⊥sta•(Y).
Conversely, let Xbe an object in ⊥sta•(Y) and ian object in Ob(I). Then for any object Yiin Yi,
one has Ext1
D-Rep(X, stai(Yi)) = 0. By Lemma 9.4 again, one concludes that Ext1
Di(coki(X), Yi) = 0,
so coki(X) is in Xi. Furthermore, since Dihas enough injectives, there exists a monomorphism
f:L1coki(X)Iiwith Iiinjective in Di. Since Ext2
Di(coki(X), Ii) = 0 = Ext1
D-Rep(X, stai(Ii))
as Ii∈Yi, it follows from Lemma 9.4 that HomDi(L1coki(X), Ii) = 0. Consequently, f= 0, so
L1coki(X) = 0, and hence X∈Φ(X). This establishes the inclusion of the other direction.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 59
The subcategory Φ(X) of D-Rep is defined by the functor cokiand its left derived functor,
which is hard to compute in practice. Our next result gives a more transparent description of this
subcategory. Recall from Definition 7.15
Φ(X) = {X∈D-Rep |ϕX
iis a monomorphism and coki(X)∈Xifor i∈Ob(I)}.
9.6 Lemma. If Iis a partially ordered category and Dis locally exact, then Φ(X) = Φ(X).
Proof. By the definitions of Φ(X) or Φ(X), to prove the conclusion, it suffices to show the following
statement: for any object Xin D-Rep and any i∈Ob(I), ϕX
iis a monomorphism if and only if
L1coki(X) = 0.
Note that we have assumed that each Dihas enough projectives at the beginning of this part.
By Proposition 5.9, D-Rep has enough projectives as well, so one has a short exact sequence
0→X′→P→X→0
in D-Rep with Pprojective. Because Dαis right exact for any α∈Pi(•, i), the above short exact
sequence induces a commutative diagram with exact rows
colimα∈Pi(•,i)Dα(X′
s(α))//
ϕX′
i
colimα∈Pi(•,i)Dα(Ps(α))//
ϕP
i
colimα∈Pi(•,i)Dα(Xs(α))//
ϕX
i
0
0//X′
i//Pi//Xi//0
in Di. By Corollary 7.19, ϕP
iis a monomorphism, and so there is an exact sequence
0→ker(ϕX
i)→coki(X′)→coki(P)→coki(X)→0
by the snake lemma. Since cokPi(P) is projective in Diby Corollary 7.19, it follows that L1coki(P) =
0. Consequently, ϕX
iis a monomorphism if and only if the sequence
0→coki(X′)→coki(P)→coki(X)→0
is exact, and if and only if L1coki(X) = 0.
The following proposition is an immediate consequence of Proposition 9.5 and Lemma 9.6.
9.7 Proposition. Suppose that Iis a partially ordered category and Dis locally exact. Then
(Φ(X),Φ(X)⊥)is a cotorsion pair in D-Rep.
The subcategory Φ(X)⊥of D-Rep still seems mysterious to us. Thus we give it a more trans-
parent interpretation under certain extra assumptions. The strategy of the proof comes from [40,
Theorem 7.9].
9.8 Proposition. Let Ibe a direct category and suppose that Dis exact. Then Φ(X)⊥=D-RepY.
Proof. Note that Φ(X) = ⊥sta•(Y) by the proof of Proposition 9.5, Φ(X) = Φ(X) by Lemma 9.6,
and D-RepY=fre•(X)⊥according to the proof of Proposition 9.3.
The inclusion Φ(X)⊥⊆D-RepYfollows from the claim fre•(X)⊆Φ(X). Indeed, if this is true,
then for an object Yin Φ(X)⊥, by Lemma 9.2 we have
Ext1
Di(Xi, Yi)∼
=Ext1
Di(Xi,evai(Y)) ∼
=Ext1
D-Rep(frei(Xi), Y ) = 0
for any i∈Ob(I) and any object Xiin Xi. Consequently, Yi∈Yisince (Xi,Yi) is a cotorsion pair
in Di. Thus Yis contained in D-RepYas desired.
To prove the above claim, we take two objects iand jin Ob(I) together with an object Xiin Xi
and an object Yjin Yj. By Lemma 9.2, we have
Ext1
D-Rep(frei(Xi),staj(Yj)) ∼
=Ext1
Di(Xi,evai(staj(Yj))).
60 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
If i=j, then evai(staj
l(Yj)) = Yi; otherwise, evai(staj(Yj)) = 0. In both cases we conclude that
Ext1
Di(Xi,evai(staj(Yj))) = 0 since (Xi,Yi) is a cotorsion pair in Di. Thus, we have
frei(Xi)∈⊥sta•(Y) = Φ(X),
so fre•(X)⊆Φ(X) as claimed.
The other inclusion D-RepY⊆Φ(X)⊥follows from the claim that every object Yin D-RepYhas
a Φ(X)⊥-cofiltration. Indeed, if this is true, then by Lemma 8.2, Y∈Φ(X)⊥as desired.
Now we prove the claim. Let {Vχ}be the transfinite sequence of subsets of Ob(I). Since Iis
a left rooted category by Proposition 7.11, Ob(I) = Vζfor a certain ordinal ζ. We construct the
following continuous inverse ζ-sequence {Yχ, gχ,χ′}χ6χ′6ζin D-Rep as in Definition 8.1.
For an ordinal χ6ζ, define Yχas follows:
•for i∈Ob(I), set
Yχ
i=(Yi,if i∈Vχ,
0,otherwise;
•for α:i→j∈Mor(I), set the structural morphism
Yχ
α:Dα(Yχ
i)→Yχ
j=(Yα,if i, j ∈Vχ,
0 otherwise.
We verify that Yχis a left representation over D. The axiom (lRep.2) holds clearly. To check
the axiom (lRep.1), we need to show
τβ,α(Yχ
i)◦Yχ
β◦α=Yχ
β◦Dβ(Yχ
α)
for any pair of composable morphisms iα
→jβ
→k∈Mor(I). If k /∈Vχ, then Yχ
βα = 0 = Yχ
β;
otherwise, if k∈Vχ, then both jand iare in Vχby Remark 7.4, and hence Yχ
βα =Yβα,Yχ
β=Yβ
and Yχ
α=Yα. In both cases the above equality holds, so Yχis indeed a left representation over D.
For ordinals χ6χ′6ζ, define gχ,χ′={gχ,χ′
i}i∈Ob(I)as follows: for any i∈Ob(I),
(a) if i∈Vχ, then i∈Vχ′(see Remark 7.2), so Yχ
i=Yiand Yχ′
i=Yi, and we set gχ,χ′
ito be idYi;
(b) if i /∈Vχ, then Yχ
i= 0, and we set gχ,χ′
ito be 0.
To verify that gχ,χ′is a morphism in D-Rep, we have to show
gχ,χ′
j◦Yχ′
α=Yχ
α◦Dα(gχ,χ′
i)
for all α:i→j∈Mor(I). If j /∈Vχ, then Yχ
j= 0; otherwise, if j∈Vχ, then both i, j ∈Vχ⊆Vχ′by
Remark 7.4, and in this case we have gχ,χ′
j= idYj,Yχ′
α=Yα,Yχ
α=Yαand Dα(gχ,χ′
i) = Dα(idYi) =
idDα(Yi). In both cases the above equality holds. Thus, gχ,χ′is a morphism in D-Rep. Furthermore,
it is easy to check that {Yχ, gχ,χ′}χ6χ′6ζis a continuous inverse ζ-sequence in D-Rep.
We finish the proof by showing that {Yχ, gχ,χ′}χ6χ′6ζis a Φ(X)⊥-cofiltration of Y(see Definition
8.1). Clearly, Y0= 0 and Yζ=Y, so it remains to show ker(gχ,χ+1)∈Φ(X)⊥for any ordinal
χ < ζ. Note that for i∈Ob(I), we have:
ker(gχ,χ+1)i=(Yi,if i∈Vχ+1 \Vχ;
0,otherwise.
Thus for a morphism α:i→j, we have two cases:
(a) if i=j, then α=eias Iis locally trivial by Proposition 7.11, so the structural morphism
ker(gχ,χ+1)eiis η−1
i(ker(gχ,χ+1)i);
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 61
(b) if i6=j, then the structural morphism
ker(gχ,χ+1)α:Dα(ker(gχ,χ+1)i)→ker(gχ,χ+1)j
is 0. Indeed, if j /∈Vχ+1\Vχ, then ker(gχ,χ+1)j= 0, so ker(gχ,χ+1)α= 0; if j∈Vχ+1\Vχ, then
i∈Vχby Remark 7.4, so ker(gχ,χ+1)i= 0 and consequently ker(gχ,χ+1)α= 0 as well.
With these observations in mind, it is routine to check that
ker(gχ,χ+1)∼
=Y
i∈Vχ+1\Vχ
stai(Yi).
But for any i∈Vχ+1\Vχ, we have
stai(Yi)∈sta•(Y)⊆(⊥sta•(Y))⊥= Φ(X)⊥.
Thus ker(gχ,χ+1)∈Φ(X)⊥as desired. This completes the proof.
The following result is a straightforward consequence of Propositions 9.7 and 9.8.
9.9 Proposition. Suppose that Iis a direct category and Dis exact. Then (Φ(X),D-RepY)is a
cotorsion pair in D-Rep.
We summarize the main results of this section in the following theorem.
9.10 Theorem. Let X={Xi⊆Di}i∈Ob(I)and Y={Yi⊆Di}i∈Ob(I)be two families of subcate-
gories such that (Xi,Yi)is a cotorsion pair in Difor each i∈Ob(I).
(a) If Dis exact, then (⊥D-RepY,D-RepY)is a cotorsion pair in D-Rep.
(b) If Iis a partially ordered category and Dis locally exact, then (Φ(X),Φ(X)⊥)is a cotorsion
pair in D-Rep.
(c) If Iis a direct category and Dis exact, then Φ(X)⊥=D-RepY, and hence (Φ(X),D-RepY)is
a cotorsion pair in D-Rep.
9.11 Remark. Let Ibe the free category associated to a quiver Qand Aa Grothendieck category
admitting enough projectives. Suppose that Dis a trivial I-diagram of the category A, that is,
Di=Afor each i∈Ob(I) and Dαis the identity functor on Afor each α∈Mor(I). The cotorsion
pairs in (a) and (b) in Theorem 9.10 were established by Holm and Jørgensen; see [40, Theorem
7.4(a)]. If furthermore Qis a left rooted quiver, then the cotorsion pair in Theorem 9.10(c) can be
found in [40, Theorem 7.9].
We end this section with the next result, which asserts that under some extra conditions the
hereditary property of the cotorsion pair (Φ(X),D-RepY) can be deduced from the hereditary prop-
erty of each cotorsion pair (Xi,Yi).
9.12 Proposition. Suppose that Iis a direct category and Dis exact. If the cotorsion pair (Xi,Yi)
is hereditary for every i∈Ob(I), then so is (Φ(X),D-RepY).
Proof. By Proposition 9.9, (Φ(X),D-RepY) is a cotorsion pair in D-Rep, so we only need to prove its
hereditary property; that is, it is both resolving and coresolving (see 8.4). However, by Theorem 2.6
and the assumption that the cotorsion pair (Xi,Yi) is coresolving, it is trivial that (Φ(X),D-RepY)
is also coresolving. Therefore, we only need to show that (Φ(X),D-RepY) is resolving.
Let 0 →M→L→K→0 be a short exact sequence in D-Rep such both Land Kare in Φ(X).
Then for i∈Ob(I), this short exact sequence induces the following commutative diagram:
0//colimα∈Pi(•,i)Dα(Ms(α))//
ϕM
i
colimα∈Pi(•,i)Dα(Ls(α))//
ϕL
i
colimα∈Pi(•,i)Dα(Ks(α))//
ϕK
i
0
0//Mi//Li//Ki//0
62 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
in Di. Note that each row in the above commutative diagram is exact. Indeed, the exactness of
the first row comes from Theorem 2.6 as well as the condition that Dαis an exact functor for any
α∈Pi(•, i). The second row is also exact by Theorem 2.6.
Since both Land Kare in Φ(X), both ϕL
iand ϕK
iare monomorphisms. Thus by the snake
lemma, ϕM
iis a monomorphism as well, and we obtain a short exact sequence
0→coker(ϕM
i)→coker(ϕL
i)→coker(ϕK
i)→0
in Di. However, by the definition of the functor coki, this short exact sequence is precisely
0→coki(M)→coki(L)→coki(K)→0.
Since both coki(L) and coki(K) are in Xiand the cotorsion pair (Xi,Yi) is resolving, coki(M) is
also in Xi. Thus Mis in Φ(X), and so (Φ(X),D-RepY) is resolving.
10. Abelian model structures on D-Rep
In this section we show that the cotorsion pair (Φ(X),D-RepY) given in Proposition 9.9 is complete
when the I-diagram Dsatisfies certain extra conditions, and hence induces an abelian model strcture
on D-Rep.
Setup. Throughout this section, let Ibe the free category associated to a left rooted quiver Q=
(Q0, Q1). Then by Theorem 1.11, there exist a strict I-diagram D′and a morphism F:D′→Dof
I-diagrams such that Fiis the identity for each i∈Ob(I), which induces an isomorphism between
the categories D′-Rep and D-Rep by Proposition 3.5. Consequently, without loss of generality, we
may suppose that Dis strict. On the other hand, by an argument similar to that Example 5.3, the
morphism category P(•, i) is a disjoint union of components each of which corresponds to a unique
arrow • → i, and this arrow is the terminal object in the component. Consequently, one has
colim
θ∈Pi(•,i)=a
θ∈Q1(•,i)
,
where Q1(•, i) is the set of arrows with target i.
10.1. Completeness of the induced cotorsion pairs. Throughout this subsection, let X=
{Xi}i∈Ob(I)and Y={Yi}i∈Ob(I)be two families of subcategories such that (Xi,Yi) is a complete
cotorsion pair in Difor each i∈Ob(I). Then we show that the cotorsion pair (Φ(X),D-RepY) given
in Proposition 9.9 is complete.
We begin with the following notion derived from some concrete cases that will be considered in
Section 11.
10.1 Definition. Let {Si}i∈Ob(I)be a family with each Sia subcategory of Di. We say that
{Si}i∈Ob(I)is compatible with respect to Dif Dα(Ss(α))⊆St(α)for any α∈Mor(I), where s(α) and
t(α) are the source and target of α, respectively.
The next result paves the way for proving the completeness of the induced cotorsion pair in
Proposition 9.9, which actually holds true for a general free category associated to a quiver.
10.2 Lemma. Suppose that Dis exact and X={Xi}i∈Ob(I)is compatible with respect to D. Then
for any object Min D-Rep, there exist short exact sequences
0→M→Y→X→0and 0→Y′→X′→M→0
in D-Rep with X, X′∈D-RepXand Y, Y ′∈D-RepY.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 63
Proof. Since (Xi,Yi) is a complete cotorsion pair for each i∈Ob(I), there exists an exact sequence
0→Mi
fi
−→ Yi
gi
−→ Xi→0
in Diwith Yi∈Yiand Xi∈Xi. Let α:i→jbe an arrow in Q1. Since Dα(Xi)⊆Xjand Dαis
exact, there exist morphisms Yαand Xαsuch that the diagram
0//Dα(Mi)Dα(fi)
//
Mα
Dα(Yi)
Yα
Dα(gi)
//Dα(Xi)
Xα
//0
0//Mj
fj
//Yj
gj
//Xj//0
commutes. By Remark 3.7, one obtains the first short exact sequence, and the second one can be
constructed dually.
Now we are able to prove the main result of this section.
10.3 Theorem. Suppose that Dis exact and X={Xi}i∈Ob(I)is compatible with respect to D.
Then (Φ(X),D-RepY)is a complete cotorsion pair in D-Rep.
Proof. It follows from Propositions 7.11 and 9.9 that (Φ(X),D-RepY) is a cotorsion pair in D-Rep,
so we only need to show its completeness. Furthermore, without loss of generality we may suppose
that Dis strict.
Let Mbe an object in D-Rep. By Lemma 10.2 we obtain a short exact sequence
E: 0 →Y→X→M→0
in D-Rep such that X∈D-RepXand Y∈D-RepY. But Xmay fail to be in Φ(X). We will use the
transfinite induction to construct another short exact sequence
E′: 0 →Y′→X′→M→0
in D-Rep with X′∈Φ(X) and Y′∈D-RepY. If this is done, the cotorsion pair (Φ(X),D-RepY) is
complete in D-Rep by Lemma 8.6(a), as D-Rep has enough (D-RepY)-objects by Lemma 10.2.
Let {Vχ}be the transfinite sequence of subsets of Ob(I) defined in 7.1. Since Iis left rooted by
Proposition 7.11, Ob(I) = Vζfor a certain ordinal ζ. Now we construct an inverse ζ-sequence
{Eχ: 0 →Yχkχ
−→ Xχhχ
−→ Mχ→0}χ6ζ
of short exact sequences in D-Rep satisfying the following conditions:
•for any ordinal 0 < χ 6ζ, one has Mχ=M;
•for any ordinal 0 < χ 6ζand any i∈Ob(I)\Vχ, one has Xχ
i=Xi∈Xiand Yχ
i=Yi∈Yi;
•for any ordinal χ6ζand any i∈Vχ,ϕXχ
iis a monomorphism, Xχ
i∈Xi,coki(Xχ)∈Xi
and Yχ
i∈Yi;
•for all 0 < χ < χ′6ζ,Xχ
i=Xχ′
iand Yχ
i=Yχ′
ifor all i /∈Vχ′\Vχ, and there exists the
following commutative diagram
Eχ′: 0 //Yχ′kχ′
//
gχ,χ′
Xχ′
fχ,χ′
hχ′
//Mχ′=M//0
Eχ: 0 //Yχkχ
//Xχhχ
//Mχ=M//0
in D-Rep such that both fχ,χ′and gχ,χ′are epimorphisms.
64 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Set E0= 0 →0→0→0→0 and E1=E. We have two cases:
Case I: χ+ 1 is a successor ordinal and we have constructed Eχ= 0 →Yχ→Xχ→M→0.
We construct Eχ+1 in the following steps.
Step 1. Construct Xχ+1 and Yχ+1.
Let i∈Vχ+1\Vχ. The cotorsion pair (Xi,Yi) is complete in Di, so there is a short exact sequence
0→a
α∈Q1(•,i)
Dα(Xχ
s(α))(σα)α∈Q1(•,i)
−−−−−−−−−→ Dχ+1
i→Xi→0
in Diwith Dχ+1
i∈Yiand Xi∈Xi. Since {Xi}i∈Ob(I)is compatible with respect to D, it follows
that `α∈Q1(•,i)Dα(Xχ
s(α)) is in Xias each α∈Q1(•, i) has the source s(α) in Vχby Remark 7.4.
Thus, Dχ+1
i∈Xi∩Yi.
For any arrow α∈Q1(•, i), there exists a canonical monomorphism
(σα)α∈Q1(•,i)◦εα:Dα(Xχ
s(α))→Dχ+1
i,
where εαis the canonical injection. On the other hand, Xχ
αis a morphism from Dα(Xχ
s(α)) to
Xχ
i=Xias i∈Ob(I)\Vχ. Thus there is a morphism
Xχ
α
(σα)α∈Q1(•,i)◦εα:Dα(Xχ
s(α))→Xi⊕Dχ+1
i
in Diwhich induces a monomorphism
(†)(Xχ
α)α∈Q1(•,i)
(σα)α∈Q1(•,i):a
α∈Q1(•,i)
Dα(Xχ
s(α))→Xi⊕Dχ+1
i,
as (σα)α∈Q1(•,i)is a monomorphism.
Now we defineXχ+1 and Yχ+1 as follows:
•For i∈Ob(I), set
Xχ+1
i=(Xχ
i,if i /∈Vχ+1\Vχ,
Xi⊕Dχ+1
i,otherwise; and Yχ+1
i=(Yχ
i,if i /∈Vχ+1\Vχ,
Yi⊕Dχ+1
i,otherwise.
Since Dχ+1
i∈Xi∩Yi, it follows that Xi⊕Dχ+1
i∈Xiand Yi⊕Dχ+1
i∈Yi. Thus Xχ+1
i∈Xi
and Yχ+1
i∈Yi.
•For an arrow γ:i→jin Q112, the structural morphisms Xχ+1
γand Yχ+1
γare defined as
follows:
(a) If j∈Vχ+1\Vχ, then i∈Vχby Remark 7.4. Define
Xχ+1
γ=Xχ
γ
(σβ)β∈Q1(•,j)◦εγ:Dγ(Xχ+1
i) = Dγ(Xχ
i)→Xχ+1
j=Xj⊕Dχ+1
j,and
Yχ+1
γ=Yχ
γ
(σβ)β∈Q1(•,j)◦εγ◦Dγ(kχ
i):Dγ(Yχ+1
i) = Dγ(Yχ
i)→Yχ+1
j=Yj⊕Dχ+1
j.
(b) If i∈Vχ+1\Vχ, then j /∈Vχ+1 by Remark 7.4 again. Thus Xχ+1
i=Xi⊕Dχ+1
i,
Xχ+1
j=Xχ
j=Xj,Yχ+1
i=Yi⊕Dχ+1
i, and Yχ+1
j=Yχ
j=Yj. Define Xχ+1
γto be
the composite of the projection Dγ(Xi⊕Dχ+1
i)∼
=Dγ(Xi)⊕Dγ(Dχ+1
i)։Dγ(Xi)
followed by Xγ:Dγ(Xi)→Xj, and define Yχ+1
γto be the composite of the projection
Dγ(Yi⊕Dχ+1
i)∼
=Dγ(Yi)⊕Dγ(Dχ+1
i)։Dγ(Yi) followed by Yγ:Dγ(Yi)→Yj.
(c) For other cases, define Xχ+1
α=Xχ
αand Yχ+1
α=Yχ
α.
12Note here that i6=jas the quiver Qis left rooted by assumption; see Lemma 7.7.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 65
By Remark 3.7, we obtain two objects Xχ+1 and Yχ+1 in D-Rep such that Xχ+1 ∈D-RepXand
Yχ+1 ∈D-RepY. To show that Xχ+1 and Yχ+1 satisfy the desired conditions (b) and (c), it remains
to prove that coki(Xχ+1 )∈Xifor all i∈Vχ+1\Vχ, that is, the cokernel of the monomorphism
ϕXχ+1
i=(Xχ
α)α∈Q1(•,i)
(σα)α∈Q1(•,i)
given in (†) is in Xi. To this end, consider the following commutative diagram
0
Xi
ι
0//`α∈Q1(•,i)Dα(Xχ
s(α))ϕXχ+1
i//Xi⊕Dχ+1
i//
π
coki(Xχ+1)//
0
0//`α∈Q1(•,i)Dα(Xχ
s(α))//Dχ+1
i
//Xi//0.
0
in Diwith exact rows and columns. Then there exists a morphism coki(Xχ+1 )→Xisuch that the
right square commutes. By the snake lemma, one gets a short exact sequence
0→Xi→coki(Xχ+1)→Xi→0
in Di. Since both Xiand Xiare in Xi, so is coki(Xχ+1 ).
Step 2. Construct Eχ+1.
Set Mχ+1 =M. Define hχ+1 :Xχ+1 →Mand kχ+1 :Yχ+1 →Xχ+1 as follows:
hχ+1
i=(hχ
i,if i /∈Vχ+1\Vχ,
(hχ
i,0),otherwise; and kχ+1
i=
kχ
i,if i /∈Vχ+1\Vχ,
kχ
i0
0 1 !,otherwise.
According to the above constructions and by Remark 3.7 again, it is routine to check that
Eχ+1 : 0 →Yχ+1 kχ+1
−−−−→ Xχ+1 hχ+1
−−−−→ Mχ+1 =M→0
is a short exact sequence in D-Rep.
Step 3. Construct epimorphisms fχ,χ+1 :Xχ+1 →Xχand gχ,χ+1 :Yχ+1 →Yχ.
Define fχ,χ+1 and gχ,χ+1 as follows:
fχ,χ+1
i=(idXχ
i,if i /∈Vχ+1\Vχ,
(1,0) ,otherwise; and gχ,χ+1
i=(idYχ
i,if i /∈Vχ+1\Vχ,
(1,0) ,otherwise.
It is clear that fχ,χ+1 and gχ,χ+1 are epimorphisms, and it is routine to check that the diagram
Eχ+1 : 0 //Yχ+1 kχ+1
//
gχ,χ+1
Xχ+1
fχ,χ+1
hχ+1
//M//0
Eχ: 0 //Yχkχ
//Xχhχ
//M//0
in D-Rep commutes.
Case II. µ6ζis a limit ordinal and Eκhas been constructed for all κ < µ. Next we construct
Eµ. In this case one has Vµ=∪κ<µVκ.
66 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
(a) If i∈Vµ, then i∈Vκfor some ordinal κ < µ, and so for all ordinal κ < κ′6µone has
Eκ′
i=Eκ
ias i /∈Vκ′\Vκ.
(b) If i /∈Vµ, then by the induction hypothesis, one has that Eκ
i=Eifor all κ < µ.
Let Eµ= limκ<µ Eκ. One has
Eµ
i= lim
κ<µ
Eκ
i=(Eκ
ifor some κ < µ if i∈Vµ,
Eiif i /∈Vµ.
Thus, Eµis an exact sequence in D-Rep satisfying the desired conditions.
In both cases we have defined Eµfor µ6ζ. In particular, for µ=ζ, one gets the desired exact
sequence E′=Eζ: 0 →Y′→X′→M→0. This finishes the proof.
10.4 Remark. In the special situation that Dis a trivial I-diagram of the category Aand (X,Y) is
a complete cotorsion pair in A, the completeness of the cotorsion pair (Φ(X),D-RepY) in Theorem
10.3 was proved by Di, Li, Liang and Xu; see [14, Theorem A and Remark 2.2]13.
The following result is an immediate consequence of Proposition 9.12 and Theorem 10.3.
10.5 Proposition. Suppose that Dis exact and X={Xi}i∈Ob(I)is compatible with respect to
D. If (Xi,Yi)is a complete and hereditary cotorsion pair in Difor every i∈Ob(I), then so is
(Φ(X),D-RepY).
10.2. Abelian model structures on D-Rep.In this subsection, as an application of Proposition
10.5, we show that under some assumptions a family {(Qi,Wi,Ri)}i∈Ob(I)of hereditary Hovey triples
on Di’s induces a hereditary Hovey triple on D-Rep.
10.6 Lemma. Suppose that S={Si}i∈Ob(I)is a family with each Sia subcategory of Diclosed
under extensions and small coproducts for each i∈Ob(I). If Sis compatible with respect to D,
then Φ(S)⊆D-RepS.
Proof. Let {Vχ}be the transfinite sequence of subsets of Ob(I) defined in 7.1. Since Iis left rooted,
Ob(I) = Vζfor a certain ordinal ζ. For each S∈Φ(S), we use the transfinite induction to show the
following assertion: for all ordinals χ, all objects i∈Vχ, one has Si∈Si.
The assertion holds trivially for V0=∅. For χ>1, we have two cases.
(a) Let χ+ 1 be a successor ordinal and i∈Vχ+1. Since S∈Φ(S), there is a short exact sequence
0→a
α∈Q1(•,i)
Dα(Ss(α))ϕS
i
−→ Si→coki(S)→0
in Diwith coki(S)∈Si. Since i∈Vχ+1, it follows from Remark 7.2 that s(α)∈Vχfor
all arrows α∈Q1(•, i). Note that {Si}i∈Ob(I)is compatible with respect to D. By the
induction hypothesis and the assumption that Siis closed under small coproducts, we deduce
that `α∈Q1(•,i)Dα(Ss(α))∈Si. But Siis closed under extensions, so Si∈Si.
(b) If χ6ζis a limit ordinal. Then the assertion is clearly true for χbecause in this case
Vχ=∪µ<χVµand the assertion holds for all ordinals µ < χ.
This yields that Sis in D-RepS, and so Φ(S)⊆D-RepS.
To describe other results, we need to fix some assumptions and notations.
Setup. Given a family {(Qi,Wi,Ri)}i∈Ob(I)with each (Qi,Wi,Ri) a Hovey triple on Di, denote by
•e
Qi(resp., e
Ri) the subcategory Qi∩Wi(resp., Ri∩Wi) for i∈Ob(I);
13In the special case where the cotorsion pair (X,Y) is assumed to be hereditary and complete, such a result was
first proved by Odaba¸sı; see [53, Theorem 4.6].
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 67
•Q(resp., W,e
Q,R,e
R) the family {Qi}i∈Ob(I)(resp., {Wi}i∈Ob(I),{e
Qi}i∈Ob(I),{Ri}i∈Ob(I),
{e
Ri}i∈Ob(I)) of subcategories.
10.7 Lemma. Suppose that Dis exact. Then any family {(Qi,Wi,Ri)}i∈Ob(I)of hereditary Hovey
triples on Disuch that both Qand e
Qare compatible with respect to Dinduces a hereditary Hovey
triple (Φ(Q),T,D-RepR)on D-Rep in which
T={X∈D-Rep |there is a s.e.s. 0→X→A→B→0with A∈D-Repe
Rand B∈Φ(e
Q)}
={X∈D-Rep |there is a s.e.s. 0→A′→B′→X→0with A′∈D-Repe
Rand B′∈Φ(e
Q)}.
Proof. By Proposition 10.5, we obtain two complete and hereditary cotorsion pairs (Φ(Q),D-Repe
R)
and (Φ(e
Q),D-RepR) in D-Rep. It is evident that Φ(e
Q)⊆Φ(Q) since e
Q⊆Q. According to Lemma
8.8, if we prove that
Φ(Q)∩D-Repe
R= Φ(e
Q)∩D-RepR,
then there exists a unique thick subcategory Twith the desired form such that (Φ(Q),T,D-RepR)
is a hereditary Hovey triple on D-Rep.
For the inclusion Φ(e
Q)∩D-RepR⊆Φ(Q)∩D-Repe
R, we take an object X∈Φ(e
Q)∩D-RepR. It
suffices to show that X∈D-Repe
R. Indeed, by Lemma 10.6, we see that Xi∈e
Qifor any i∈Ob(I).
Thus Xi∈e
Qi∩Ri=Qi∩Wi∩Ri⊆e
Ri, so X∈D-Repe
Ras desired.
For the other inclusion, let Xbe an object in Φ(Q)∩D-Repe
R. By Lemma 10.6 again, we have
Xi∈Qi∩e
Ri=e
Qi∩Ri, so it remains to prove that coki(X) in the exact sequence
0→a
α∈Q1(•,i)
Dα(Xs(α))ϕX
i
−−→ Xi→coki(X)→0
belongs to e
Qi=Qi∩Wi. Since coki(X)∈Qi, it remains to show coki(X)∈Wi. But Wiis a thick
subcategory of Di, so it is enough to show that the first two terms are in Wi. This is clearly true
for Xias it belongs to e
Ri=Wi∩Ri.
We finish the proof by showing that the first term lies in Wi. Since Qs(α)∩e
Rs(α)=e
Qs(α)∩Rs(α)
for any arrow α∈Pi(•, i), we conclude that Xs(α)∈e
Qs(α). But {e
Qi}i∈Ob(I)is compatible with
respect to Dby assumption, so Dα(Xs(α))∈e
Qi. Consequently, `α∈Q1(•,i)Dα(Xs(α))∈e
Qias e
Qiis
closed under small coproducts. In particular, `α∈Q1(•,i)Dα(Xs(α))∈Wias e
Qi=Qi∩Wi.
The next result shows that the subcategory Tof trivial objects in the Hovey triple given in
Lemma 10.7 coincides with the subcategory D-RepW.
10.8 Lemma. Keep the assumptions and notations as in Lemma 10.7. Then T=D-RepW.
Proof. The inclusion T⊆D-RepWholds as Φ(e
Q)⊆D-Repe
Qby Lemma 10.6, so we only need to
show the inclusion D-RepW⊆T.
Let Xbe an object in D-RepW. For any i∈Ob(I), since (Qi,e
Ri) is a complete cotorsion pair
in Diand {Qi}i∈Ob(I)is compatible with respect to Dby assumption, it follows from Lemma 10.2
that there exists a short exact sequence
0→X→A→B→0
in D-Rep with A∈D-Repe
Rand B∈D-RepQ. Since both Xiand Aibelong to Wi, which is a thick
subcategory of Di, we conclude that Bi∈Wias well, so B∈D-Repe
Q. But Bmight not be in Φ(e
Q).
We will use transfinite induction to construct an exact sequence
E′: 0 →X→A′→B′→0
68 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
in D-Rep such that A′∈D-Repe
Rand B′∈Φ(e
Q). If this is done, then Xlies in T.
Let {Vχ}be the transfinite sequence of subsets of Ob(I) defined in 7.1. Since Iis left rooted,
Ob(I) = Vζfor some ordinal ζ. We construct an inverse ζ-sequence
{Eχ: 0 →Xχkχ
−→ Aχhχ
−→ Bχ→0}χ6ζ
of short exact sequences in D-Rep satisfying the following conditions:
•for ordinal 0 < χ 6ζ,Xχ=X;
•for ordinal 0 < χ 6ζand object i∈Ob(I)\Vχ,Aχ
i=Ai∈e
Riand Bχ
i=Bi∈e
Qi;
•for ordinal χ6ζand object i∈Vχ,Aχ
i∈e
Ri,ϕBχ
iis a monomorphism, Bχ
i∈e
Qi, and
coki(Bχ)∈e
Qi;
•for ordinals 0 < χ < χ′6ζ,Aχ
i=Aχ′
iand Bχ
i=Bχ′
ifor objects i /∈Vχ′\Vχ, and there
exists a commutative diagram
Eχ: 0 //Xχkχ
//Aχ
fχ′,χ
hχ
//Bχ
gχ′,χ
//0
Eχ′: 0 //Xχ′kχ′
//Aχ′hχ′
//Bχ′
//0
in D-Rep such that both fχ′,χ and gχ′,χ are epimorphisms.
Now we can just mimic the construction in the proof of Theorem 10.3 to do that. Finally we
obtain the desired exact sequence E′=Eζ: 0 →X→A′→B′→0.
The main result of this section is as follows, which is immediate from Lemmas 10.7 and 10.8.
10.9 Theorem. Suppose that Dis exact. Then a family {(Qi,Wi,Ri)}i∈Ob(I)of hereditary Hovey
triples on Disuch that Qand e
Qare compatible with respect to Dinduces a hereditary Hover triple
(Φ(Q),D-RepW,D-RepR)on D-Rep.
10.10 Remark. This result has been proved in [13, Theorem B] for the category of representations
from a left rooted quiver Qto a Grothendieck category admitting enough projectives, which is a
special case of the above theorem by taking Dto be the corresponding trivial diagram. We shall
also point out that for an I-diagram Dof model categories where Iis a direct category, Greenlees
and Shipley constructed a diagram-projective model structure on D-Rep; see [36, Theorem 3.1].
They did this by constructing a weak equivalence and a (co)fibration in D-Rep, while we construct
model structures through the cotorsion theory and Hovey correspondence.
10.11 Remark. All results in Sections 9 and 10 have their dual versions on the subcategory Ψ(Y)
of the right representation category Rep-Dintroduced in Definition 7.21.
11. Applications to diagrams of module categories
Let Rbe an I-diagram of associative rings. In this section we describe some applications of the
main results in previous sections to representations over I-diagrams Rand Rof module categories
induced by R. More specifically, under some assumptions, we give characterizations of (Gorenstein)
flat objects in R-Rep as well as characterizations of Gorenstein injective objects in Rep-R, and
establish a Gorenstein flat model structure on R-Rep as well as a Gorenstein injective model
structure on Rep-R.
Let us recall some results established in previous sections. According to Subsection 4.1, Rinduces
a right exact endo-trivial I-diagram Rof module categories with
•Ri=Ri-Mod for i∈Ob(I) and
•Rα=Rj⊗Ri−:Ri-Mod →Rj-Mod for α:i→j∈Mor(I),
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 69
and a left exact endo-trivial I-diagram Rof module categories with
•Ri=Mod-Rifor i∈Ob(I) and
•Rα= HomRi(Rj,−) : Mod-Ri→Mod-Rjfor α:i→j∈Mor(I).
It is clear that Rα(resp., Rα) preserves small coproducts (resp., products) for any α∈Mor(I).
By Proposition 4.5, the category R-Mod of left R-modules (resp., the category Mod-Rof right R-
modules ) introduced in [26, Definition 3.6] coincides with the category R-Rep (resp., the category
Rep-R).
11.1. Flat objects. In this subsection we give a description of the categorical flat objects in R-Rep
based on the the categorical tensor product due to Oberst and Rohrl [52]. We recall the following
construction.
Given an object Min R-Rep and a Z-module G, we define an object Hom(M , G) in Rep-Ras
follows:
•for i∈Ob(I), set Hom(M, G)ito be HomZ(Mi, G)∈Mod-Ri;
•for α:i→j∈Mor(I), there exists a morphism Mα:Rα(Mi) = Rj⊗RiMi→Mjin Rj,
which yields a morphism Hom(M , G)j= HomZ(Mj, G)HomZ(Mα,G)
//HomZ(Rj⊗RiMi, G)
in Rj. Set Hom(M , G)αto be the composite of HomZ(Mα, G) and the classical isomorphism
Isoα: HomZ(Rj⊗RiMi, G)→HomRi(Rj,HomZ(Mi, G)) = Rα(Hom(M, G)i).
It is routine to check that Hom(M, G) satisfies the axioms (rRep.1) and (rRep.2) in Definition
2.13, and hence is an object in Rep-R. It is also clear that Hom(M , −) is a functor from Z-Mod
to Rep-R, and is left exact and preserves products. Consequently, it has a left adjoint, denoted by
− ⊗RM:Rep-R→Z-Mod, and called the tensor product functor.
From the definition we immediately obtain:
11.1 Lemma. Let Mbe an object in R-Rep and Nan object in Rep-R. Then for any Z-module
G, there exists a natural isomorphism
HomZ(N⊗RM , G)∼
=HomRep-R(N, Hom(M, G)).
Now we can define categorical flat objects in R-Rep.
11.2 Definition. An object Fin R-Rep is called flat if the functor − ⊗RFis exact. The subcat-
egory of all flat objects in R-Rep is denoted by Flat(R-Rep).
An equivalent characterization of categorical flat object is:
11.3 Lemma. An object Fin R-Rep is flat if and only if F+= HomZ(F, Q/Z)is injective in
Rep-R.
Proof. Given any short exact sequence 0 →N→N′→N′′ →0 in Rep-R, we consider the
following commutative diagram
0//HomRep-R(N, F +)//
∼
=
HomRep-R(N′, F +)//
∼
=
HomRep-R(N′′, F +)//
∼
=
0
0//(N⊗RF)+//(N′⊗RF)+//(N′′ ⊗RF)+//0
of abelian groups, where the columns are isomorphisms by Lemma 11.1. Then the first row is exact
if and only if so is the second one, or equivalently, Fis flat if and only if F+is injective.
From now on, we denote by Flat•the family {Flat(Ri-Mod)}i∈Ob(I)of subcategories consisting of
flat objects in Ri-Mod.
70 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
11.4 Lemma. Suppose that Iis a partially ordered category. Then an object Min R-Rep is
contained in the subcategory Φ(Flat•)of R-Rep if and only if M+is contained in the subcategory
Ψ(Inj•)of Rep-R.
Proof. The statement that M∈Φ(Flat•) is equivalent to the second statement: for all j∈Ob(I)
there exists a short exact
0→colim
σ∈Pj(•,j)(Rj⊗Rs(σ)Fs(σ))ϕF
j
−→ Fj→cokj(F)→0
in Rj-Mod with cokj(F) flat; see Definition 7.15. This turns out to be equivalent to the third
statement: for j∈Ob(I), there exists a short exact sequence
0→(cokj(F))+→(Fj)+(ϕF
j)+
−→ ( colim
σ∈Pj(•,j)(Rj⊗Rs(α)Fs(α)))+→0
in Mod-Rjwith (cokj(F))+injective. However, since
( colim
σ∈Pj(•,j)(Rj⊗Rs(σ)Fs(σ)))+= HomZ( colim
σ∈Pj(•,j)(Rj⊗Rs(σ)Fs(σ)),Q/Z)
∼
=lim
σ∈Pj(•,j)HomZ(Rj⊗Rs(σ)Fs(σ),Q/Z)
∼
=lim
σ∈Pj(•,j)HomRs(σ)(Rj,HomZ(Fs(σ),Q/Z))
= lim
σ∈Pj(•,j)HomRs(σ)(Rj,(Fs(σ))+),
and (ϕF
j)+is nothing but ψF+
jby definitions, the third statement is equivalent to the statement
that M+∈Ψ(Inj•); see Subsection 6.3.
As an immediate consequence of Corollaries 7.19 and 7.23 and Lemmas 11.3 and 11.4, we obtain
the following characterization of flat objects in R-Rep.
11.5 Theorem. Suppose that Iis a direct category. Then Flat(R-Rep) = Φ(Flat•).
11.6 Remark. Theorem 11.5 was first proved by Enochs, Oyonarte and Torrecillas in [23, Theorem
3.7] for the trivial diagram indexed by the free category associated to a left rooted quiver.
11.2. Gorenstein injective model structures. In this subsection we consider Gorenstein injec-
tive objects in Rep-R, and construct Gorenstein injective model structures on Rep-R.
Enochs and Jenda introduced in [21] the notion of Gorenstein injective modules over associative
rings. Analogously, we give the following definition of Gorenstein injective objects in Rep-R.
11.7 Definition. An object Nin Rep-Ris called Gorenstein injective if there is an exact sequence
I:· · · → I−1→I0→I1→ · · ·
in Rep-Rwith Ijinjective for all j∈Zsuch that N∼
=ker (I0→I1) and the sequence Iremains
exact after applying the functor HomRep-R(E, −) for every injective object Ein Rep-R. The
subcategory of all Gorenstein injective objects in Rep-Ris denoted by GI(Rep-R).
Denote by GI•the family {GI(Mod-Ri)}i∈Ob(I)of subcategories consisting of Gorenstein injective
objects in Mod-Ri. Our next task is to show that under some assumptions the subcategories
GI(Rep-R) and Ψ(GI•) of Rep-Rcoincide. Before proving this result, we need to finish a few
preparatory works.
Denote by Proj•(resp., Inj•) the family {Proj(Ri-Mod)}i∈Ob(I)(resp., {Inj(Mod-Ri)}i∈Ob(I)) of
subcategories of Ri-Mod (resp., Mod-Ri) consisting of projective objects (resp., injective objects).
It is easy to show the following result.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 71
11.8 Lemma. The family Proj•is compatible with respect to Rand the family Inj•is compatible
with respect to R.
Recall that the I-diagram Rof associative rings is called flat if the functor Rαis flat for any
α:i→j∈Mor(I), that is, the tensor product functor Rj⊗Ri−:Ri-Mod →Rj-Mod is exact. As
we described at the beginning of Section 10, one has
lim
θ∈Pi(•,i)=Y
θ∈Q1(•,i)
whenever Iis the free category associated to a quiver Q= (Q0, Q1).
11.9 Lemma. Suppose that Iis the free category associated to a left rooted quiver Q, and Ris
flat. Let Nbe a Gorenstein injective object in Rep-R. Then for any i∈Ob(I), one has:
(a) ψN
i:Ni→Qθ∈Q1(•,i)HomRs(θ)(Ri, Ns(θ))is an epimorphism;
(b) keri(N) = ker(ψN
i)is Gorenstein injective in Mod-Ri.
That is, GI(Rep-R)⊆Ψ(GI•).
Proof. Fix an object i∈Ob(I) and consider the exact sequence of injective objects in Definition
11.7. For all integers j, since Ij∈Ψ(Inj•) by Corollary 7.23, there exists a short exact sequence
0→keri(Ij)→Ij
i→Y
θ∈Q1(•,i)
HomRs(θ)(Ri, Ij
s(θ))→0
in Mod-Riwith keri(Ij) injective. For any arrow θ∈Q1(•, i), we have an exact sequence
Is(θ)=· · · → I−1
s(θ)→I0
s(θ)→I1
s(θ)→ · · ·
in Mod-Rs(θ). Since the family Inj•is compatible with respect to Rby Lemma 11.8, we conclude
by the dual version of Lemma 10.6 that Ij
s(θ)is injective in Mod-Rs(θ). Hence, it follows from
[63, Corollary 5.9] that the sequence HomRs(θ)(Ri,Is(θ)) is exact as Riis flat in Mod-Rs(θ). This
implies that the sequence Qθ∈Q1(•,i)HomRs(θ)(Ri,Is(θ)) is also exact. Consequently, we obtain the
commutative diagram
.
.
.
.
.
.
.
.
.
0//keri(I−1)
//I−1
i
//Qθ∈Q1(•,i)HomRs(θ)(Ri, I−1
s(θ))
//0
0//keri(I0)
//I0
i
//Qθ∈Q1(•,i)HomRs(θ)(Ri, I0
s(θ))
//0
0//keri(I1)
//I1
i
//Qθ∈Q1(•,i)HomRs(θ)(Ri, I1
s(θ))
//0
.
.
..
.
..
.
.
with exact rows and columns, which induces the short exact sequence
0→keri(N)→Ni
ψN
i
−→ Y
θ∈Q1(•,i)
HomRs(θ)(Ri, Ns(θ))→0.
Therefore, to complete the proof, it remains to show that keri(N) is Gorenstein injective in Mod-Ri,
that is, the sequence HomRi(Ei,keri(I)) is exact for any injective object Eiin Mod-Ri. Indeed, we
72 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
have
HomRep-R(frei(Ei),I)∼
=HomRep-R(frei(Ei),Y
i∈Ob(I)
frei(keri(I)))
∼
=Y
i∈Ob(I)
HomRep-R(frei(Ei),frei(keri(I)))
∼
=Y
i∈Ob(I)
HomRi(evai(frei(Ei)),keri(I)),
where the first isomorphism holds by Remark 7.25 and the third isomorphism holds as (evai,frei)
is an adjoint pair by Corollary 5.19. Since frei(Ei) is injective in Rep-Ras the functor evaiis
exact, the sequence HomRep-R(frei(Ei),I) is exact by Definition 11.7. Consequently, the sequence
HomRi(evai(frei(Ei)),keri(I)) is also exact by the above isomorphisms. The conclusion then follows
by observing that evai(frei(Ei)) = Rei(Ei) = HomRi(Ri, Ei)∼
=Ei.
11.10 Lemma. Suppose that Ris flat and satisfies one of the following two conditions:
(a) Riis commutative for all i∈Ob(I);
(b) Rjhas finite projective dimension as a left Ri-module for all α:i→j∈Mor(I).
Then the family GI•is compatible with respect to R.
Proof. In both cases Rα= HomRi(Rj, Ei) is Gorenstein injective in Mod-Rjfor any α:i→j∈
Mor(I) and any Gorenstein injective object Eiin Mod-Ri; see [11, Lemma 3.4] and [10, Ascent
table II(h)]. The conclusion then follows.
Denote by ⊥GI•the family {⊥GI(Mod-Ri)}i∈Ob(I)of subcategories of Mod-Ri, and by (Rep-R)⊥GI•
the subcategory of Rep-Rconsisting of objects Xwith Xicontained in ⊥GI(Mod-Ri) for i∈Ob(I).
By [62, Theorem 5.6], (⊥GI(Mod-Ri),GI(Mod-Ri)) is a complete hereditary cotorsion pair in Mod-Ri.
Now we describe a cotorsion pair in Rep-Rsuch that Ψ(GI•) is the right half of this pair.
11.11 Proposition. Suppose that Iis a direct category and Ris flat. Then ((Rep-R)⊥GI•,Ψ(GI•))
is a cotorsion pair in Rep-R. If Iis the free category associated to a left rooted quiver and Rsatisfies
further the condition (a) or (b) in Lemma 11.10, then the cotorsion pair ((Rep-R)⊥GI•,Ψ(GI•)) is
complete and hereditary.
Proof. Take an object i∈Ob(I), an object N′
i∈GI(Mod-Ri) and an object N∈Rep-R. Consider
a short exact sequence 0 →N′
i→Ki→evai(N)→0 in Mod-Ri. Since Ris flat, Rjis a flat object
in Mod-Rifor any α:i→j∈Mor(I). It follows from [63, Corollary 5.9] that Ext1
Ri(Rj, N ′
i) = 0.
Thus the above short sequence remains exact after applying the functor frei. Since (evai,frei) is an
adjoint pair by Corollary 5.19, we have an isomorphism
Ext1
Rep-R(N, frei(N′
i)) ∼
=Ext1
Mod-Ri(evai(N), N ′
i)
by Lemma 9.1. Now by an argument similar to the one obtaining Proposition 9.9, one can show
that ((Rep-R)⊥GI•,Ψ(GI•)) is a cotorsion pair in Rep-R. Moreover, note that the family GI•is
compatible with respect to Rby Lemma 11.10. By an argument similar to the one obtaining
Proposition 10.5, one can show that the cotortion pair is complete and hereditary.
11.12 Remark. The above result is not a special case of the dual version of Proposition 10.5,
though it is proved in a similar way. Indeed, even if Ris flat, the I-diagram Rof module categories
is not exact in general, that is, Rα= HomRi(Rj,−) might not be exact for every α:i→j∈Mor(I).
The key point we used here is that Ext1
Ri(Rj, Ni) = 0 for each α:i→j∈Mor(I) and each
Gorenstein injective right Ri-module Ni; see [63, Corollary 5.9].
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 73
11.13 Proposition. Suppose that Iis the free category associated to a left rooted quiver, and R
is flat and satisfies condition (a) or (b) in Lemma 11.10. Then there is a hereditary Hovey triple
(Rep-R,(Rep-R)⊥GI•,Ψ(GI•))
on Rep-R.
Proof. Note that (Mod-Ri,⊥GI(Mod-Ri),GI(Mod-Ri)) is a hereditary Hovey triple on Mod-Ri;
see [62, Page 31]. By Lemmas 11.8 and 11.10, both families Inj•and GI•are compatible with respect
to R. The conclusion then follows by an argument similar to the one obtaining Theorem 10.9; here
we do not assume that Rα= HomRi(Rj,−) is exact.
We recall the following definition from Auslander and Buchweitz [4] .
11.14 Definition. Let Ybe a subcategory of an abelian category A. A subcategory Vof Yis called
agenerator for Yif for each object Y∈Ythere exists a short exact sequence 0 →Y′→V→Y→0
with V∈Vand Y′∈Y. Dually, Vis a cogenerator for Yif there exists a short exact sequence
0→Y→V→Y′→0 with V∈Vand Y′∈Y.
11.15 Lemma. Suppose that Iis the free category associated to a left rooted quiver, and Ris flat
and satisfies condition (a) or (b) in Lemma 11.10. Then Ψ(Inj•)is a generator for Ψ(GI•).
Proof. Take an object Nin Ψ(GI•). Since ((Rep-R)⊥GI•,Ψ(GI•)) is a complete cotorsion pair in
Rep-Rby Proposition 11.11, there exists a short exact sequence 0 →N′→E→N→0 in Rep-R
with E∈(Rep-R)⊥GI•∩Ψ(GI•) and N′∈Ψ(GI•). We finish the proof by showing E∈Ψ(Inj•).
Indeed, by Proposition 11.13 and Corollary 7.23, one has
(Rep-R)⊥GI•∩Ψ(GI•) = Inj(Rep-R) = Ψ(Inj•).
This finishes the proof.
Now we are ready to characterize Gorenstein injective objects in Rep-Rfor certain special dia-
grams of module categories.
11.16 Theorem. Suppose that Iis the free category associated to a left rooted quiver, and Ris
flat and satisfies the condition (a) or (b) in Lemma 11.10. Then GI(Rep-R) = Ψ(GI•).
Proof. The inclusion GI(Rep-R)⊆Ψ(GI•) holds by Lemma 11.9. For the other inclusion, we take
an object N∈Ψ(GI•) and show that N∈GI(Rep-R).
Since Inj(Rep-R) is a generator for Ψ(GI•) by Corollary 7.23 and Lemma 11.15, there exists a short
exact sequence 0 →N−1→I−1→N→0 in Rep-Rwith I−1∈Inj(Rep-R) and N−1∈Ψ(GI•). For
any injective object Ein Rep-R, since the family Inj•is compatible with respect to Rby Lemma
11.8, we conclude by the dual version of Lemma 10.6 that E∈(Rep-R)Inj•⊆(Rep-R)⊥GI•. Thus
by Proposition 11.11, Ext1
Rep-R(E, N −1) = 0, so the above short exact sequence remains exact after
applying the functor HomRep-R(E, −). Continuing this process for N−1, eventually one gets an
exact sequence
· · · → I−2→I−1→N→0 (†)
in Rep-Rwith Ij∈Inj(Rep-R) for all integers j < 0 such that the sequence (†) remains exact after
applying the functor HomRep-R(E, −).
On the other hand, since Rep-Rhas enough injectives by Proposition 5.20, we get an exact
sequence
0→N→I0→I1→ · · · (‡)
in Rep-Rwith Ij∈Inj(Rep-R) for all integers j≥0. By Proposition 11.11 again, Ψ(GI•) is closed
under taking cokernels of monomorphisms, so all cokernels of the sequence (‡) belong to Ψ(GI•)
by noting that each Ij∈Ψ(Inj•) and Ψ(Inj•)⊆Ψ(GI•). Using the same argument as above, we
conclude that the sequence (‡) remains exact after applying the functor HomRep-R(E, −).
74 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Assembling the sequences (†) and (‡), we deduce that N∈GI(Rep-R), as desired.
11.17 Remark. The above theorem generalizes [24, Theorem 3.5.1] by Eshraghi, Hafezi and
Salarian, where they proved the same result for the trivial diagram.
As an immediate consequence of Proposition 11.11 and Theorem 11.16, we have:
11.18 Corollary. Suppose that Iis the free category associated to a left rooted quiver, and R
is flat and satisfies the condition (a) or (b) in Lemma 11.10. Then ((Rep-R)⊥GI•,GI(Rep-R)) is a
complete and hereditary cotorsion pair in Rep-R.
The following result gives a Grenstein injective model structure on Rep-R.
11.19 Corollary. Suppose that Iis the free category associated to a left rooted quiver, and Ris
flat and satisfies the condition (a) or (b) in Lemma 11.10. Then there is a hereditary Hovey triple
(Rep-R,⊥GI(Rep-R),GI(Rep-R))
on Rep-R.
Proof. By Propositin 11.13, there exists a hereditary Hovey triple
(Rep-R,(Rep-R)⊥GI•,Ψ(GI•))
on Rep-R. The conclusion then follows from Theorem 11.16 and Corollary 11.18.
11.3. Gorenstein flat model structures. In this subsection we consider Gorenstein flat objects
in R-Rep, which have certain special flat property with respect to the categorical tensor product
introduced previously, and construct Gorenstein flat model structures on R-Rep.
Enochs, Jenda and Torrecillas introduced in [23] the notion of Gorenstein flat modules over asso-
ciative rings. Projectively coresolved Gorenstein flat modules over associative rings were introduced
by ˇ
Saroch and ˇ
St’ov´ıˇcek in [62]. We extend them to representations over diagrams.
11.20 Definition. An object Min R-Rep is called Gorenstein flat if there is an exact sequence
F:· · · → F−1→F0→F1→ · · ·
in R-Rep where Fjis flat for every j∈Zsuch that M∼
=ker (F0→F1) and the sequence remains
exact after applying the functor E⊗R−for every injective object Ein Rep-R.
Similarly, one can define projectively coresolved Gorenstein flat representations when flat repre-
sentations in the above exact sequence are replaced by projective representations.
We introduce a few more notations. Denote by
•GF(Ri-Mod) the subcategory of Gorenstein flat objects in Ri-Mod;
•PGF(Ri-Mod) the subcategory of projectively coresolved Gorenstein flat objects in Ri-Mod;
•GF•the family {GF(Ri-Mod)}i∈Ob(I)of subcategories of Ri-Mod;
•PGF•the family {PGF(Ri-Mod)}i∈Ob(I)of subcategories of Ri-Mod;
•PGF⊥
•the family {PGF(Ri-Mod)⊥}i∈Ob(I)of subcategories of Ri-Mod;
•GF(R-Rep) the subcategory of R-Rep consisting of Gorenstein flat objects;
•PGF(R-Rep) the subcategory of R-Rep consisting of projectively coresolved Gorenstein flat
objects.
Recall that an object Miin Ri-Mod is called cotorsion if Ext1
Ri(Fi, Mi) = 0 for any flat object
Fiin Ri-Mod. Similarly, one can define cotorsion objects in R-Rep. Denote by
•Cot(Ri-Mod) the subcategory of cotorsion objects in Ri-Mod;
•Cot•the family {Cot(Ri-Mod)}i∈Ob(I)of subcategories of Ri-Mod;
•Cot(R-Rep) the subcategory of R-Rep consisting of cotorsion objects.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 75
In the following lemma we collect some elementary properties of Gorenstein flat objects in R-Rep,
which are quite similar to those of Gorenstein injective objects in Rep-R; see Lemma 11.9.
11.21 Lemma. Suppose that Iis the free category associated to a left rooted quiver Q, and Ris
flat. Let Mbe an Gorenstein flat object in R-Rep. Then for any i∈Ob(I), one has:
(a) φM
i:`θ∈Q1(•,i)(Ri⊗Rs(θ)Ms(θ))→Miis a monomorphism, and
(b) coki(M) = coker(φM
i)is Gorenstein flat in Ri-Mod.
That is, GF(R-Rep)⊆Φ(GF•).
Proof. Fix an object i∈Ob(I) and consider the exact sequence Fof flat objects in Definition
11.20. For all integers j, since Fj∈Φ(Flat•) by Theorem 11.5, there is a short exact sequence
0→a
θ∈Q1(•,i)
(Ri⊗Rs(θ)Fj
s(θ))→Fj
i→coki(Fj)→0
in Ri-Mod with coki(Fj) flat. For any arrow θ∈Q1(•, i), we have an exact sequence
Fs(θ):· · · → F−1
s(θ)→F0
s(θ)→F1
s(θ)→ · · ·
in Rs(θ). Since Riis flat in Mod-Rs(θ)as Ris flat, the sequence Ri⊗Rs(θ)Fs(θ)is exact. This implies
that the sequence `θ∈Pi(•,i)(Ri⊗Rs(θ)Fs(θ)) is exact. Consequently, we obtain the commutative
diagram
.
.
.
.
.
.
.
.
.
0//`θ∈Q1(•,i)(Ri⊗Rs(θ)F−1
s(θ))
//F−1
i
//coki(F−1)
//0
0//`θ∈Q1(•,i)(Ri⊗Rs(θ)F0
s(θ))
//F0
i
//coki(F0)
//0
0//`θ∈Q1(•,i)(Ri⊗Rs(θ)F1
s(θ))
//F1
i
//coki(F1)
//0
.
.
..
.
..
.
.
with exact rows and columns, which induces the short exact sequence
0→a
θ∈Q1(•,i)
(Ri⊗Rs(θ)Ms(θ))φM
i
−→ Mi→coki(M)→0.
Therefore, to complete the proof, it remains to show that coki(M) is Gorenstein flat in Ri-Mod,
that is, the sequence Ii⊗Ricoki(F) is exact for any injective object Iiin Mod-Ri.
For any injective object E∈Rep-R, the sequence HomRep-R(E, F+) is exact as
HomRep-R(E, F+)∼
=(E⊗RF)+
by Lemma 11.1. By an argument similar to the one used in the proof of Lemma 11.9, we con-
clude that the sequence HomRi(Ii,keri(F+)) is exact. Now the conclusion follows from the above
isomorphism and the observation that keri(F+) is exactly coki(F)+.
11.22 Lemma. If Ris flat, then the families GF•and PGF•are compatible with respect to R.
76 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Proof. We only deal with GF•as the argument also works for PGF•with small modifications. Since
Ris flat, Rjis flat in Ri-Mod for any α:i→j∈Mor(I). By [10, Ascent table II(a)], Rj⊗RiGi
is Gorenstein flat in Rj-Mod for any Gorenstein flat object Giin Ri-Mod. Thus the family GF•is
compatible with respect to R.
By [62, Corollary 4.12], (GF(Ri-Mod),GF(Ri-Mod)⊥) is a complete and hereditary cotorsion
pair in Ri-Mod for i∈Ob(I). Moreover, by [62, Theorem 4.9], (PGF(Ri-Mod),PGF(Ri-Mod)⊥) is
another complete and hereditary cotorsion pair in Ri-Mod. Thus by Lemma 11.22 and Proposition
10.5, we obtain the following result.
11.23 Proposition. Suppose that Iis the free category associated to a left rooted quiver and R
is flat. Then:
(a) (Φ(GF•),R-RepGF⊥
•)is a complete and hereditary cotorsion pair in R-Rep.
(b) (Φ(PGF•),R-RepPGF⊥
•)is a complete and hereditary cotorsion pair in R-Rep.
The complete hereditary cotorsion pairs in the above proposition induce two hereditary Hovey
triples. Explicitly, one has:
11.24 Proposition. Suppose that Iis the free category associated to a left rooted quiver and R
is flat. Then there exist hereditary Hovey triples
(Φ(GF•),R-RepPGF⊥
•,R-RepCot•)and (Φ(PGF•),R-RepPGF⊥
•,R-Rep)
on R-Rep.
Proof. There are two hereditary Hovey triples
(GF(Ri-Mod),PGF(Ri-Mod)⊥,Cot(Ri-Mod)) and (PGF(Ri-Mod),PGF(Ri-Mod)⊥, Ri-Mod)
on Ri-Mod; see [62, Page 27]. Moreover, the families GF•,PGF•and Proj•are compatible with
respect to Rby Lemmas 11.22 and 11.8, and clearly Flat•is also compatible with respect to R.
Thus by Theorem 10.9, we obtain the desired hereditary Hovey triples on R-Rep.
Next we show that under some conditions the subcategories Cot(R-Rep) and R-RepCot•coincide.
11.25 Proposition. Suppose that Iis the free category associated to a left rooted quiver Q, and
Ris flat. Then Cot(R-Rep) = R-RepCot•.
Proof. Since (Flat(Ri-Mod),Cot(Ri-Mod)) is a cotorsion pair in Ri-Mod for each i∈Ob(I), it
follows from Proposition 9.9 that (Φ(Flat•),R-RepCot•) is a cotorsion pair in R-Rep. On the other
hand, note that the notion of flat objects given in Definition 11.2 is indeed the categorical flat objects
in R-Rep; see [52]. It follows that (Flat(R-Rep),Cot(R-Rep)) is also a cotorsion pair in R-Rep.
However, by Theorem 11.5, we have Flat(R-Rep) = Φ(Flat•), so Cot(R-Rep) = R-RepCot•.
The next result will be applied in the proof of Theorem 11.28.
11.26 Lemma. Suppose that Iis the free category associated to a left rooted quiver and Ris flat.
Then the following statements hold.
(a) Φ(Flat•)is a cogenerator for Φ(GF•).
(b) Φ(Proj•)is a cogenerator for Φ(PGF•).
Proof. We only prove the statement (a); the statement (b) is proved similarly.
Let Mbe an object in Φ(GF•). Since (Φ(GF•),R-RepGF⊥
•) is a complete cotorsion pair in R-Rep
by Proposition 11.23(a), there exists a short exact sequence 0 →M→F→M′→0 in R-Rep
with F∈Φ(GF•)∩R-RepGF⊥
•and M′∈Φ(GF•). Thus it is enough to show F∈Φ(Flat•).
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 77
By Proposition 11.24, (Φ(GF•),R-RepPGF⊥
•,R-RepCot•) is a Hovey triple on R-Rep, so the pair
(Φ(GF•),R-RepPGF⊥
•∩R-RepCot•) is a cotorsion pair. Thus by Proposition 11.23(a), we have
R-RepGF⊥
•=R-RepPGF⊥
•∩R-RepCot•.
We also have Φ(GF•)∩R-RepPGF⊥
•=Flat(R-Rep) since (Flat(R-Rep),Cot(R-Rep)) is a cotorsion
pair in R-Rep and Cot(R-Rep) = R-RepCot•by Proposition 11.25. It follows immediately that
Φ(GF•)∩R-RepGF⊥
•=Φ(GF•)∩R-RepPGF⊥
•∩R-RepCot•⊆Flat(R-Rep).
However, by Theorem 11.5, Flat(R-Rep) = Φ(Flat•), so Φ(GF•)∩R-RepGF⊥
•⊆Φ(Flat•).Thus we
have F∈Φ(Flat•).
According to [39, Theorem 3.6], if Miis Gorenstein flat in Ri-Mod, then M+
iis Gorenstein
injective in Mod-Ri. The converse statement holds true whenever Riis right coherent. Relying on
this fact, one can prove the following result by an argument similar to the one used in the proof of
Lemma 11.4.
11.27 Lemma. Suppose that Iis a partially ordered category and let Mbe an object in R-Rep.
If Mis contained in Φ(GF•), then M+is contained in Ψ(GI•). Furthermore, the converse statement
holds if Riis right coherent for every i∈Ob(I).
Now we can give characterizations of Gorenstein flat objects and projectively coresolved Goren-
stein flat objects in R-Rep for some special kind of diagrams Rof module categories.
11.28 Theorem. Suppose that Iis the free category associated to a left rooted quiver, and Ris
flat and satisfies condition (a) or (b) in Lemma 11.10. Then
GF(R-Rep) = Φ(GF•)and PGF(R-Rep) = Φ(PGF•).
Proof. We only show the first equality since the second one can be proved similarly. Lemma 11.21
tells us that GF(R-Rep)⊆Φ(GF•), so we only need to show the other inclusion.
Take an object M∈Φ(GF•). By Theorem 11.5 and Lemma 11.26(a), Φ(Flat•) = Flat(R-Rep) is
a cogenerator for Φ(GF•), so there is a short exact sequence 0 →M→F0→M1→0 in R-Rep
with F0∈Flat(R-Rep) and M1∈Φ(GF•), which induces a short exact sequence
0→(M1)+→(F0)+→M+→0
in Rep-R. By Lemma 11.27, (M1)+is contained in Ψ(GI). Therefore, by Theorem 11.16, (M1)+is
Gorenstein injective in Rep-R, and hence the sequence
0→HomRep-R(E, (M1)+)→HomRep-R(E, (F0)+)→HomRep-R(E, M +)→0
is exact for any injective object E∈Rep-R. It follows from Lemma 11.1 that the sequence
0→E⊗RM→E⊗RF0→E⊗RM1→0
is also exact. Replacing Mby M1, recursively one gets an exact sequence
0→M→F0→F1→ · · · (†)
in R-Rep with Fj∈Flat(R-Rep) for all integers j≥0 such that the sequence (†) remains exact
after applying the functor E⊗R−.
On the other hand, note that R-Rep has enough projectives by Proposition 5.9, so there exists
an exact sequence
· · · → F−2→F−1→M→0 (‡)
in R-Rep with Fj∈Proj(R-Rep)⊆Flat(R-Rep) for all integers j < 0. Since Φ(GF•) is closed
under taking kernels of epimorphisms by Proposition 11.23, all kernels of the sequence (‡) belong
78 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
to Φ(GF•) by noting that each Fj∈Φ(Flat•) and Φ(Flat•)⊆Φ(GF•) clearly. Using a similar
argument as before we get that the sequence (‡) remains exact after applying the functor E⊗R−.
Assembling the sequences (†) and (‡), we deduce that Mis in GF(R-Rep), as desired.
11.29 Remark. Theorem 11.28 was first proved by Di, Estrada, Liang and Odaba¸sı in [13, Theo-
rem A and Theorem 3.13] for the trivial diagram.
As an immediate consequence of Proposition 11.23 and Theorem 11.28, we have:
11.30 Corollary. Suppose that Iis the free category associated to a left rooted quiver, and
Ris flat satisfying condition (a) or (b) in Lemma 11.10. Then (GF(R-Rep),R-RepGF⊥
•)and
(PGF(R-Rep),R-RepPGF⊥
•)are complete and hereditary cotorsion pairs in R-Rep.
We end this section with the following result which provides a Gorenstein flat model structure
on R-Rep.
11.31 Corollary. Suppose that Iis the free category associated to a left rooted quiver, and Ris
flat satisfying condition (a) or (b) in Lemma 11.10. Then there is a hereditary Hovey triple
(GF(R-Rep),PGF(R-Rep)⊥,Cot(R-Rep))
on R-Rep.
Proof. By Proposition 11.24, there is a hereditary Hovey triple (Φ(GF),R-RepPGF⊥
•,R-RepCot•) on
R-Rep. However, note that Φ(GF•) = GF(R-Rep) by Theorem 11.28, R-RepPGF⊥
•=PGF(R-Rep)⊥
by Corollary 11.30 and R-RepCot•=Cot(R-Rep) by Proposition 11.25. The conclusion follows.
11.32 Remark. Corollary 11.31 was first proved by Di, Estrada, Liang and Odaba¸sı in [13, The-
orem C] for the trivial diagram.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 79
Appendix A: Diagrams of small preadditive categories
We have claimed that all relevant results in Section 4 can extend from I-diagrams of associative
rings to I-diagrams of small preadditive categories. In this appendix we mainly show that an I-
diagram Rof small preadditive categories induces two diagrams R:I→AB and R:I→AB. The
processes become a bit more complicated (but not very difficult) in this more general case. For the
reader’s convenience, we recall some necessary definitions and results appeared in [26, Section 2].
A.1 Modules over small preadditive categories. Let Sbe a small preadditive category. Recall
that a left (resp., right)S-module Mis an additive functor M:S→Ab (resp., M:Sop →Ab), and
a homomorphism φ:M→Nof left S-modules is a natural transformation consisting of a family
{φ(s) : M(s)→N(s)}s∈Ob(S)of homomorphisms of abelian groups such that the square
M(s)φ(s)
//
M(α)
N(s)
N(α)
M(t)φ(t)
//N(t)
commutes for any morphism α:s→tin Mor(S). Similarly, one can give the definition of ho-
momorphisms of right S-modules. Denote by (S, Ab) (resp., (Sop,Ab)) the category of left (resp.,
right) S-modules.
The following result is well-known.
A.2 Lemma. Let Sbe a small preadditive category. Then the category (S, Ab) (resp., (Sop ,Ab)) of
left (resp., right)S-modules is a Grothendieck category with a family of small projective generators.
A.3 Tensor product of small preadditive categories. Let Sand Tbe small preadditive
categories. Recall that the tensor product S⊗Tis defined as follows:
•Ob(S⊗T) = Ob(S)×Ob(T);
•given two objects (s1, t1) and (s2, t2) in Ob(S⊗T),
HomS⊗T((s1, t1),(s2, t2)) = HomS(s1, s2)⊗ZHomT(t1, t2);
•composition of morphisms in S⊗Tis defined by
(f1⊗g1)◦(f2⊗g2) = (f1◦f2)⊗(g1◦g2).
By properties of tensor products of abelian groups, it is not difficult to verify that S⊗Tis a
small preadditive category, and the operation to form tensor products is associative.
A.4 Bimodules. Recall that a (T , S)-bimodule Mis an additive functor M:Sop ⊗T→Ab.For
example, HomS(−,−): Sop ⊗S→Ab is an (S, S )-bimodule.
Given a (T, S )-bimodule M:Sop ⊗T→Ab, one obtains a right S-module M(−, t): Sop →Ab
for any t∈Ob(T) and a left T-module M(s, −) : T→Ab for any s∈Ob(S). In particular, given
an additive functor φ:S→T, one obtains a (T , S)-bimodule HomT(φ(−),−): Sop ⊗T→Ab.
A.5 Tensor product of modules. Let Mbe right S-module and Na left S-module. The tensor
product of Mand Nis defined as
M⊗SN=
M
s∈Ob(S)
M(s)⊗ZN(s)
/∼,
where ∼is the subgroup generated by elements of the form M(α)(x)⊗y−x⊗N(α)(y) with
α:a→b∈Mor(S), x∈M(b) and y∈N(a).
80 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Given homomorphisms φ:M→M′in (Sop,Ab) and ψ:N→N′in (S, Ab), the homomorphism
φ⊗Sψ:M⊗SN→M′⊗SN′
of abelian groups is defined as the morphism induced on the quotient by the diagonal morphism
M
s∈Ob(S)
φ(s)⊗Zψ(s): M
s∈Ob(S)
M(s)⊗ZN(s)→M
s∈Ob(S)
M′(s)⊗ZN′(s).
The following well known result gives some natural properties of tensor products of modules; see
for instance [26, Lemma 2.7].
A.6 Lemma. Let Sbe a small preadditive category. Then the following statements hold.
(a) The tensor product −⊗S−: (Sop ,Ab)×(S, Ab)→Ab commutes with colimits in both variables.
(b) HomS(−, s)⊗SM∼
=M(s)for all s∈Ob(S)and all left S-modules M.
(c) N⊗SHomS(s, −)∼
=N(s)for all s∈Ob(S)and all right S-modules N.
A.7 Tensor product of bimodules. Let S,Tand Ube small preadditive categories. Following
[26, Definition 2.9], one defines a bifunctor
− ⊗T−: (Top ⊗U, Ab)×(Sop ⊗T, Ab)→(Sop ⊗U, Ab)
as follows:
•Given a (U, T )-bimodule Mand a (T , S)-bimodule N, define M⊗TN:Sop ⊗U→Ab via
letting (M⊗TN)(s, u) = M(−, u)⊗TN(s, −) for u∈Ob(U) and s∈Ob(S).
•Given morphisms α:s1→s2∈Mor(Sop) and β:u1→u2∈Mor(U),
(M⊗TN)(α⊗β): (M⊗TN)(s1, u1)−→ (M⊗TN)(s2, u2)
is defined as the tensor product over Tof the two morphisms Mβ:M(−, u1)→M(−, u2)
of right T-modules and Nα:N(s1,−)→N(s2,−) of left T-modules.
•Given homomorphisms φ:M→M′of (U, T )-bimodules and ψ:N→N′of (T , S)-
bimodules, the homomorphism φ⊗ψ:M⊗TM′→N⊗TN′of (U, S)-bimodules is defined
as (φ⊗ψ)(s, u) = φ(−, u)⊗ψ(s, −).
It is routine to verify that the above definition indeed gives a bifunctor. Furthermore, the
following result [26, Lemma 2.10] shows that the tensor product of bimodules is associative.
A.8 Lemma. Let S,T,Uand Vbe small preadditive categories, and let M:Top ⊗S→Ab,
N:Uop ⊗T→Ab and K:Vop ⊗U→Ab be bimodules. Then there is a natural isomorphism
M⊗T(N⊗UK)→(M⊗TN)⊗UK
of (S, V )-bimodules such that the component (M⊗T(N⊗UK))(v, s)→((M⊗TN)⊗UK)(v, s)
with s∈Ob(S)and v∈Ob(V)is determined by the assignment l⊗(h⊗k)7→ (l⊗h)⊗kfor
l∈M(t, s),h∈N(u, t)and k∈K(v, u).
A.9 Change of base. Let φ:S→Tbe an additive functor between small preadditive categories.
Recall from [26, Subsection 2.3] that the restriction of scalars along φis the functor
φ∗= (− ◦ φ): (T , Ab)→(S, Ab)
which is defined by composing a functor N:T→Ab with φto obtain a functor
φ∗(N) = N◦φ:S→Ab.
The extension of scalars along φis the functor
φ!= HomT(φ(−),−)⊗S−: (S, Ab)→(T, Ab)
which is the tensor product over Sby the (T, S )-bimodule HomT(φ(−),−).
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 81
The extension φ!of scalars is the left adjoint of the restriction φ∗of scalars, and we say that the
adjunction (φ!, φ∗) is the change of base adjunction along φ. It is clear that φ∗is exact and φ!is
only right exact in general.
Given additive functors φ:S→Tand ψ:T→Ubetween small preadditive categories, it is
easy to check that φ∗◦ψ∗= (ψ◦φ)∗, so we obtain a natural isomorphism ψ!◦φ!∼
=(ψ◦φ)!, whose
explicit description is given in the proof of the following lemma ([26, Lemma 2.15]).
A.10 Lemma. Let φ:S→Tand ψ:T→Ube two additive functors between small preadditive
categories. Then there exists a natural isomorphism
ζψ,φ : HomU(ψ(−),−)⊗T(HomT(φ(−),−)⊗S−)−→ HomU(ψ(φ(−)),−)⊗S−.
Proof. For any left S-module M, define
ζψ,φ(M) : HomU(ψ(−),−)⊗T(HomT(φ(−),−)⊗SM)−→ HomU(ψ(φ(−)),−)⊗SM
by composing the natural isomorphism
HomU(ψ(−),−)⊗T(HomT(φ(−),−)⊗SM)−→ (HomU(ψ(−),−)⊗THomT(φ(−),−)) ⊗SM
described in Lemma A.8 with the natural isomorphism
HomU(ψ(−), u)⊗THomT(φ(s),−)∼
−→ HomU(ψ(φ(s)), u),
determined by the assignment
(ψ(t)g
−→ u)⊗(φ(s)f
−→ t)7−→ (ψ(φ(s)) g◦ψ(f)
−→ u)
with t∈Ob(T), s∈Ob(S) and u∈Ob(U). More explicitly, the map ζψ,φ(M) is defined by the
assignments
(ψ(t)g
−→ u)⊗((φ(s)f
−→ t)⊗m)7−→ (ψ(φ(s)) g◦ψ(f)
−→ u)⊗m,
and the inverse map ζ−1
ψ,φ(M) is defined by the assignement
(ψ(φ(s)) h
−→ u)⊗m7−→ h⊗(idφ(s)⊗m)
for all m∈M.
Now we show that an I-diagram Rof small preadditive categories induces a diagram (R, η, τ ) :
I→AB of abelian categories.
A.11 Example. Let (R, η, τ ) be an I-diagram of small preadditive categories. Define (R, η, τ ) :
I→AB as follows:
•For i∈Ob(I), set Rito be the abelian category (Ri,Ab).
•For α:i→jin Mor(I), set Rαto be the functor
(Rα)!= HomRj(Rα(−),−)⊗Ri−: (Ri,Ab)→(Rj,Ab).
•Fix a left Ri-module M∈(Ri,Ab). Note that for a∈Ob(Ri), there exists an isomorphism
θa:M(a)→HomRi(−, a)⊗RiM= HomRi(idRi(−), a)⊗RiM
of abelian groups by Lemma A.6(b). This yields a natural isomorphism
θ:M→HomRi(idRi(−),−)⊗RiM
of left Ri-modules. On the other hand, since ηi: idRi→Reiis a natural isomorphism by
Definition 1.1, there exists an isomorphism
HomRi(η−1
i(−), a)⊗RiidM: HomRi(idRi(−), a)⊗RiM→HomRi(Rei(−), a)⊗RiM
of abelian groups, which yields a natural isomorphism
HomRi(η−1
i(−),−)⊗RiidM: HomRi(idRi(−),−)⊗RiM→HomRi(Rei(−),−)⊗RiM
82 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
of left Ri-modules. Consequently, one gets a natural isomorphism
(HomRi(η−1
i(−),−)⊗RiidM)◦θ:M→HomRi(Rei(−),−)⊗RiM
of left Ri-modules, which is defined to be ηi(M). In this way we obtain
ηi: id(Ri,Ab)
≃
−→ (Rei)!=Rei.
•For any pair of composable morphisms α:i→jand β:j→kin Mor(I), by Lemma A.10,
there exists a natural isomorphism
ζRβ,Rα: (Rβ)!◦(Rα)!= HomRk(Rβ(−),−)⊗Rj(HomRj(Rα(−),−)⊗Ri−)−→
(Rβ◦Rα)!= HomRk(Rβ(Rα(−)),−)⊗Ri−;
see the proof of Lemma A.10 for details. On the other hand, since τβ,α :Rβ◦Rα→Rβ◦α
is a natural isomorphism by Definition 1.1, there exists a natural isomorphism
(τ−1
β,α)!: (Rβ◦Rα)!→(Rβ◦α)!= HomRk(Rβ◦α(−),−)⊗Ri−.
Thus one obtains a natural isomorphism
(τ−1
β,α)!◦ζRβ,Rα:Rβ◦Rα→Rβ◦α
which is defined to be τβ,α .
In the rest of this appendix, for any pair of composable morphisms αand βin Mor(I), denote
the natural isomorphism ζRβ,Rαby ζβ,α for brevity.
A.12 Rsatisfies the axioms (Dia.1) and (Dia.2). We prove that (R, η, τ ) defined above does
satisfy the axioms (Dia.1) and (Dia.2) in Definition 1.1, so it is indeed an I-diagram of abelian
categories. Furthermore, it is right exact since Rα= HomRj(Rα(−),−)⊗Ri−is a right exact
functor for all α∈Mor(I).
For the axiom (Dia.1), let iα
→jβ
→kγ
→lbe three composable morphisms in Mor(I). We claim
that the natural transformation
(τ−1
γ,β ∗idRα)!: (Rγ◦Rβ◦Rα)!= HomRl(Rγ(Rβ(Rα(−))),−)⊗Ri− −→
(Rγ◦β◦Rα)!= HomRl(Rγ◦β(Rα(−)),−)⊗Ri−
between two functors from (Ri,Ab) to (Rl,Ab) (see A.9) can be explicitly described as
ζγβ,α ◦((τ−1
γ,β )!∗id(Rα)!)◦(ζγ,β ∗id(Rα)!)◦(id(Rγ)!∗ζ−1
β,α)◦(id(Rγ)!∗(τβ,α)!)◦ζ−1
γ,βα ◦(idRγ∗τ−1
β,α)!,
which is equivalent to the equality
(τ−1
γ,β ∗idRα)!(M)(b) = ζγβ,α (M)(b)◦((τ−1
γ,β )!∗id(Rα)!)(M)(b)◦(ζγ,β ∗id(Rα)!)(M)(b)
◦(id(Rγ)!∗ζ−1
β,α)(M)(b)◦(id(Rγ)!∗(τβ,α)!)(M)(b)◦ζ−1
γ,βα(M)(b)◦(idRγ∗τ−1
β,α)!(M)(b)
for M∈(Ri,Ab) and b∈Ob(Rl), since the naturality holds clearly. Note that both are maps from
M
a∈Ob(Ri)
HomRl(Rγ(Rβ(Rα(a))), b)⊗ZM(a)
/∼
to the abelian group
M
a∈Ob(Ri)
HomRl(Rγ◦β(Rα(a)), b)⊗ZM(a)
/∼;
see A.5. Thus we only need to show that they send the image in the quotient of any object f⊗m
with f∈HomRl(Rγ(Rβ(Rα(a))), b) and m∈M(a) to the same target.
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 83
On one hand, we have
[(τ−1
γ,β ∗idRα)!(M)(b)](f⊗m) = (f◦τ−1
γ,β (Rα(a))) ⊗m.
On the other hand, by the definitions of Godement products (see Definition 1.0.1), ζ−1
•,•(M) and
ζ•,•(M) (see the proof of Lemma A.10), one has the following equalities
{ζγβ,α(M)(b)◦((τ−1
γ,β )!∗id(Rα)!)(M)(b)◦(ζγ,β ∗id(Rα)!)(M)(b)◦(id(Rγ)!∗ζ−1
β,α)(M)(b)
◦(id(Rγ)!∗(τβ,α)!)(M)(b)◦ζ−1
γ,βα(M)(b)}[((idRγ∗τ−1
β,α)!(M)(b))(f⊗m)]
={ζγβ,α(M)(b)◦((τ−1
γ,β )!∗id(Rα)!)(M)(b)◦(ζγ,β ∗id(Rα)!)(M)(b)◦(id(Rγ)!∗ζ−1
β,α)(M)(b)
◦(id(Rγ)!∗(τβ,α)!)(M)(b)}[(ζ−1
γ,βα(M)(b))((f◦Rγ(τ−1
β,α(a))) ⊗m)]
={ζγβ,α(M)(b)◦((τ−1
γ,β )!∗id(Rα)!)(M)(b)◦(ζγ,β ∗id(Rα)!)(M)(b)◦(id(Rγ)!∗ζ−1
β,α)(M)(b)}
[((id(Rγ)!∗(τβ,α)!)(M)(b))((f◦Rγ(τ−1
β,α(a))) ⊗(idRβα (a)⊗m))]
={ζγβ,α(M)(b)◦((τ−1
γ,β )!∗id(Rα)!)(M)(b)◦(ζγ,β ∗id(Rα)!)(M)(b)}[((id(Rγ)!∗ζ−1
β,α)(M)(b))
((f◦Rγ(τ−1
β,α(a))) ⊗((idRβα (a)◦τβ,α(a)) ⊗m))]
={ζγβ,α(M)(b)◦((τ−1
γ,β )!∗id(Rα)!)(M)(b)}[((ζγ,β ∗id(Rα)!)(M)(b))
((f◦(Rγ(τ−1
β,α(a)))) ⊗((idRβα (a)◦τβ,α(a)) ⊗(idRα(a)⊗m)))]
={ζγβ,α(M)(b)}[((τ−1
γ,β )!∗id(Rα)!)(M)(b)((f◦Rγ(τ−1
β,α (a)) ◦Rγ(τβ,α (a))) ⊗(idRα(a)⊗m))]
={ζγβ,α(M)(b)}[((τ−1
γ,β )!∗id(Rα)!)(M)(b)(f⊗(idRα(a)⊗m))]
= (ζγβ,α(M)(b))((f◦τ−1
γ,β (Rα(a))) ⊗(idRα(a)⊗m))
= (f◦τ−1
γ,β (Rα(a)) ◦Rγ β (idRα(a))) ⊗m
= (f◦τ−1
γ,β (Rα(a))) ⊗m.
Thus the above two maps are identical, and we have the following equalities
τγ,βα ◦(idRγ∗τβ,α )
=τγ,βα ◦(id(Rγ)!∗τβ,α)
= (τ−1
γ,βα)!◦ζγ ,βα ◦(id(Rγ)!∗((τ−1
β,α)!◦ζβ,α ))
= ((idRγ∗τβ,α)◦(τ−1
γ,β ∗idRα)◦τ−1
γβ,α)!◦ζγ ,βα ◦(id(Rγ)!∗((τ−1
β,α )!◦ζβ,α ))
= (τ−1
γβ,α)!◦(τ−1
γ,β ∗idRα)!◦(idRγ∗τβ,α)!◦ζγ ,βα ◦(id(Rγ)!∗((τ−1
β,α)!◦ζβ,α ))
= (τ−1
γβ,α)!◦(τ−1
γ,β ∗idRα)!◦(idRγ∗τβ,α)!◦ζγ ,βα ◦(id(Rγ)!∗(τ−1
β,α)!)◦(id(Rγ)!∗ζβ,α )
= (τ−1
γβ,α)!◦ζγ β,α ◦((τ−1
γ,β )!∗id(Rα)!)◦(ζγ,β ∗id(Rα)!)◦(id(Rγ)!∗ζ−1
β,α)◦(id(Rγ)!∗(τβ,α )!)◦ζ−1
γ,βα◦
(idRγ∗τ−1
β,α)!◦(idRγ∗τβ,α)!◦ζγ,β α ◦(id(Rγ)!∗(τ−1
β,α)!)◦(id(Rγ)!∗ζβ,α)
= (τ−1
γβ,α)!◦ζγ β,α ◦((τ−1
γ,β )!∗id(Rα)!)◦(ζγ,β ∗id(Rα)!)
= (τ−1
γβ,α)!◦ζγ β,α ◦(((τ−1
γ,β )!◦ζγ,β )) ∗id(Rα)!)
=τγβ,α ◦(τγ,β ∗id(Rα)!)
=τγβ,α ◦(τγ,β ∗idRα),
where the first holds by the definition of Rγ, the second holds by the definitions of τγ,βα and τβ ,α,
the third holds by the axiom (Dia.1) in Definition 1.1, the fifth and eighth hold by Lemma 1.10,
the sixth equality holds by the identity we just established, the ninth holds by the definitions of
τγβ,α and τγ,β , and the last one holds by the definition of Rα. Thus Rsatisfies the axioms (Dia.1).
84 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
For the axiom (Dia.2), take iα
→j∈Mor(I). We want to show
idRα=τα,ei◦(idRα∗ηi) = (ηj∗idRα)◦τej,α,
where Rα= (Rα)!= HomRj(Rα(−),−)⊗Ri−is a functor (Ri,Ab)→(Rj,Ab); see Example A.11
for details. We prove the first identity as the second one can be established in a similar way. It
suffices to show (τα,ei◦(id(Rα)!∗ηi))(M)(c), an endomorphism of the abelian group
M
a∈Ob(Ri)
HomRj(Rα(a), c)⊗ZM(a)
/∼,
is actually the identity for M∈(Ri,Ab) and c∈Ob(Rj), that is, it sends the image in the quotient
of f⊗mwith f∈HomRj(Rα(a), c) and m∈M(a) to itself. Indeed, one has the following equalities
[(τα,ei◦(id(Rα)!∗ηi))(M)(c)](f⊗m)
= [τα,ei(M)(c)◦(id(Rα)!∗ηi)(M)(c)](f⊗m)
= [(τ−1
α,ei)!(M)(c)◦ζα,ei(M)(c)◦(id(Rα)!∗ηi)(M)(c)](f⊗m)
={(τ−1
α,ei)!(M)(c)◦ζα,ei(M)(c)}[((id(Rα)!∗ηi)(M)(c))(f⊗m)]
={(τ−1
α,ei)!(M)(c)}[(ζα,ei(M)(c))(f⊗(η−1
i(a)⊗m))]
= [(τ−1
α,ei)!(M)(c)](f◦(Rα(η−1
i(a)⊗m)))
= (f◦τ−1
α,ei(a)◦Rα(η−1
i(a))) ⊗m
= (f◦τ−1
α,ei(a)◦(idRα∗η−1
i)(a)) ⊗m
= (f◦idRα(a)) ⊗m
=f⊗m,
where the second holds by the definition of τα,ei(see Example A.11), the fourth holds by the
definitions of Godement product and ηi(see (1.0.1) and Example A.11), the fifth holds by the
definition of ζα,ei(M) (see the proof of Lemma A.10), the seventh holds by (1.0.1), and the eighth
holds by the axiom (Dia.2) in Definition 1.1. Thus Rsatisfies the axiom (Dia.2).
We end this appendix with the following example showing that an I-diagram Rof small pread-
ditive categories also induces a diagram (R, η, τ ) : I→AB.
A.13 Example. Let (R, η, τ ) be an I-diagram of small preadditive categories. Define (R, η, τ ) :
I→AB as follows:
•For i∈Ob(I), set Ri= (Rop
i,Ab), the category of right Ri-modules.
•For α:i→jin Mor(I), since HomRj(Rα(−),−) is an (Rj, Ri)-bimodule, we set Rαto be
the functor HomRi(HomRj(Rα(−),−),−) : (Rop
i,Ab)→(Rop
j,Ab).
•For i∈Ob(I), take a right Ri-module Nin (Rop
i,Ab). Since there exists an isomorphism
ξa:N(a)→HomRi(HomRi(−, a), N ) = HomRi(HomRi(idRi(−), a), N )
of abelian groups for all a∈Ob(Ri) by Yoneda’s Lemma, one gets a natural isomorphism
ξ:N→HomRi(HomRi(idRi(−),−), N)
of right Ri-modules. On the other hand, since ηi: idRi→Reiis a natural isomorphism by
Definition 1.1, there exists an isomorphism
HomRi(HomRi(ηi(−), a), N) : HomRi(HomRi(idRi(−), a), N)→HomRi(HomRi(Rei(−), a), N )
of abelian groups, which yields a natural isomorphism
HomRi(HomRi(ηi(−),−), N) : HomRi(HomRi(idRi(−),−), N )→HomRi(HomRi(Rei(−),−), N)
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 85
of right Ri-modules. Thus we obtain a natural isomorphism
HomRi(HomRi(ηi(−),−), N)◦ξ:N→HomRi(HomRi(Rei(−),−), N )
of right Ri-modules, which is defined to be ηi(N). This gives the definition of
ηi: id(Rop
i,Ab)
≃
−→ HomRi(HomRj(Rei(−),−),−) = Rei.
•For a pair of composable morphisms α:i→jand β:j→kin Mor(I), take a right
Ri-module Nin (Rop
i,Ab). By the adjunction, there exists a natural isomorphism ϑ(N) of
right Rk-modules from HomRj(HomRk(Rβ(−),−),HomRi(HomRj(Rα(−),−), N )) to
HomRi(HomRk(Rβ(−),−)⊗RjHomRj(Rα(−),−), N).
Since there exists an isomorphism
Rα(−): HomRk(Rβ(−),−)⊗RjHomRj(Rα(−),−)→HomRk(Rβ(Rα(−)),−)
of abelian groups by Lemma A.6(c), one gets a natural isomorphism HomRi(Rα(−), N ) of
right Rk-modules from HomRi(HomRk(Rβ(−),−)⊗RjHomRj(Rα(−),−), N ) to
HomRi(HomRk(Rβ(Rα(−)),−), N).
On the other hand, note that τβ,α :Rβ◦Rα→Rβα is a natural isomorphism by Defini-
tion 1.1, so there exists a natural isomorphism HomRi(HomRi(τβ,α(−),−), N ) of right Rk-
modules from HomRi(HomRk(Rβ(Rα(−)),−), N ) to HomRi(HomRk(Rβ◦α(−),−), N). Thus
one obtains an isomorphism HomRi(HomRi(τβ,α (−),−), N )◦HomRi(Rα(−), N)◦ϑ(N) of
right Rk-modules from HomRj(HomRk(Rβ(−),−),HomRi(HomRj(Rα(−),−), N )) to
HomRi(HomRk(Rβα(−),−), N ).
Finally, we get an isomorphism HomRi(HomRi(τβ ,α(−),−),−)◦HomRi(Rα(−),−)◦ϑ(−)
from the functor Rβ◦Rα= HomRj(HomRk(Rβ(−),−),HomRi(HomRj(Rα(−),−),−)) to
the functor Rβα = HomRi(HomRk(Rβα(−),−),−) , which is defined to be τβ,α.
One can show that (R, η, τ ) satisfies the axioms (Dia.1) and (Dia.2). Therefore, it is an I-diagram
of abelian categories. Furthermore, since the functor Rα= HomRi(HomRj(Rα(−),−),−) is clearly
left exact for all α∈Mor(I), the I-diagram (R, η, τ ) is left exact.
Appendix B: Details in proofs
In this appendix, we give details omitted in the proofs.
86 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
B.1 Details in the proof of Proposition 1.11. For case (c) in the proof of Proposition 1.11,
we have:
τβ,α ◦(Fβ∗Fα) = τβ,α ◦(idDβ∗Fα)
=τβ,α ◦(idDβ∗((ταn,αn−1◦···◦α1)◦(idDαn∗ταn−1,αn−2◦···◦α1)◦
· · · ◦ (idDαn◦···◦Dα3∗τα2,α1)))
=τβ,α ◦(
n−1
z}| {
(idDβ◦ · · · ◦ idDβ)∗((ταn,αn−1◦···◦α1)◦(idDαn∗ταn−1,αn−2◦···◦α1)◦
· · · ◦ (idDαn◦···◦Dα3∗τα2,α1)))
=τβ,α ◦(idDβ∗ταn,αn−1◦···◦α1)◦(idDβ∗(idDαn∗ταn−1,αn−2◦···◦α1))◦
· · · ◦ (idDβ∗(idDαn◦···◦Dα3∗τα2,α1))
=τβ,α ◦(idDβ∗ταn,αn−1◦···◦α1)◦((idDβ∗idDαn)∗ταn−1,αn−2◦···◦α1)◦
· · · ◦ ((idDβ∗idDαn◦···◦Dα3)∗τα2,α1))
=τβ,α ◦(idDβ∗ταn,αn−1◦···◦α1)◦(idDβ◦Dαn∗ταn−1,αn−2◦···◦α1)◦
· · · ◦ (idDβ◦Dαn◦···◦Dα3∗τα2,α1)
=Fβ◦αn◦···◦α1=Fβα,
where the fourth equality holds by Lemma 1.10, and the fifth equality follows from the associativity
of the operation “∗”.
For case (d), we have:
τβ,α ◦(Fβ∗Fα)
=τβ,α ◦((τβm,β′◦(Fβm∗Fβ′)) ∗Fα) by case (c)
=τβ,α ◦((τβm,β′◦(idDβm∗Fβ′)) ∗Fα) by def. of Fβm
=τβ,α ◦((τβm,β′◦(idDβm∗Fβ′)) ∗(idDα◦Fα))
=τβ,α ◦(τβm,β′∗idDα)◦((idDβm∗Fβ′)∗Fα) by Lemma 1.10
=τβ,α ◦(τβm,β′∗idDα)◦(idDβm∗(Fβ′∗Fα)) by associativity of ∗
=τβ,α ◦(τβm,β′∗idDα)◦(idDβm∗(τ−1
β′,α ◦Fβ′α)) by induction assumption
=τβ,α ◦(τβm,β′∗idDα)◦((idDβm◦idDβm)∗(τ−1
β′,α ◦Fβ′α))
=τβ,α ◦(τβm,β′∗idDα)◦(idDβm∗τ−1
β′,α)◦(idDβm∗Fβ′α) by Lemma 1.10
=τβ,α ◦(τβm,β′∗idDα)◦(idDβm∗τβ′,α)−1◦(idDβm∗Fβ′α)
=τβm,β′α◦(idDβm∗Fβ′α) by (Dia.1)
=Fβα.by case (c)
B.2 Details in the proof of Theorem 2.6. We check that Cok(ω) is an object in D-Rep. Given
two composable morphisms α:i→jand β:j→kin Mor(I), we have the following equalities
Cok(ω)β◦Dβ(Cok(ω)α)◦τ−1
β,α(Cok(ω)i)◦Dβα(πi)
= Cok(ω)β◦Dβ(Cok(ω)α)◦((Dβ◦Dα)(πi)) ◦τ−1
β,α(M′
i)
= Cok(ω)β◦Dβ(πj)◦Dβ(M′
α)◦τ−1
β,α(M′
i) by diagram (2.6.1)
=πk◦M′
β◦Dβ(M′
α)◦τ−1
β,α(M′
i) by diagram (2.6.1)
=πk◦M′
βα by (lRep.1)
= Cok(ω)βα ◦Dβα (πi),by diagram (2.6.1)
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 87
where the first equality holds by the commutative diagram
Dβα(M′
i)
τ−1
β,α (M′
i)
Dβα (πi)
//Dβα(Cok(ω)i)
τ−1
β,α (Cok(ω)i)
(Dβ◦Dα)(M′
i)(Dβ◦Dα)(πi)
//(Dβ◦Dα)(Cok(ω)i)
induced by applying the natural isomorphism τ−1
β,α to the morphism πi. So one has
Cok(ω)β◦Dβ(Cok(ω)α)◦τ−1
β,α(Cok(ω)i) = Cok(ω)βα ,
as Dβα(πi) is an epimorphism. That is, Cok(ω) satisfies the axiom (lRep.1).
For i∈Ob(I), we have the following equalities
Cok(ω)ei◦ηi(Cok(ω)i)◦πi
= Cok(ω)ei◦Dei(πi)◦ηi(M′
i)
=πi◦M′
ei◦ηi(M′
i) by diagram (2.6.1)
=πi◦idM′
iby (lRep.2)
= idCok(ω)i◦πi,
where the first equality holds by the commutative diagram induced by applying the natural iso-
morphism ηito the morphism πi. Consequently,
Cok(ω)ei◦ηi(Cok(ω)i) = idCok(ω)i,
as πiis an epimorphism, so Cok(ω) satisfies the axiom (lRep.2).
Now we check the equality
M′′
α◦Dα(ρi) = ρj◦Cok(ω)α.(‡)
Note that
M′′
α◦Dα(ρi)◦Dα(πi) = M′′
α◦Dα(ρi◦πi)
=M′′
α◦Dα(ǫi) by diagram (2.6.2)
=ǫj◦M′
α
=ρj◦πj◦M′
αby diagram (2.6.2)
=ρj◦Cok(ω)α◦Dα(πi).by diagram (2.6.1),
Since Dα(πi) is an epimorphism, the desired equality follows.
B.3 Details in the proof of Proposition 3.1. In this subsection we verify that F!constructed
before Proposition 3.1 is indeed a functor by showing that it sends objects to objects and morphisms
to morphisms.
88 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
To verify that F!(N) satisfies the axiom (lRep.1), let α:i→jand β:j→kbe two composable
morphisms in Mor(I). Then we have
F!(N)βα
=Fk(Nβα)◦F−1
βα (Ni) by def. of F!(N)βα
=Fk(Nβ)◦Fk(D′
β(Nα)) ◦Fk(τ′−1
β,α(Ni)) ◦F−1
βα (Ni) by (lRep.1)
=Fk(Nβ)◦Fk(D′
β(Nα)) ◦Fk(τ′−1
β,α(Ni)) ◦(idFk∗τ′
β,α)(Ni)◦
(F−1
β∗idD′
α)(Ni)◦(idDβ∗F−1
α)(Ni)◦(τ−1
β,α ∗idFi)(Ni) by (Mor.1)
=Fk(Nβ)◦Fk(D′
β(Nα)) ◦Fk(τ′−1
β,α(Ni)) ◦idFk(D′
βα(Ni)) ◦Fk(τ′
β,α(Ni))◦
F−1
β(D′
α(Ni)) ◦Dβ(Fj(idD′
α(Ni))) ◦idDβ(Fj(D′
α(Ni))) ◦Dβ(F−1
α(Ni))◦
τ−1
β,α(Fi(Ni)) ◦Fi(idFi(Ni)) by def. of ∗
=Fk(Nβ)◦Fk(D′
β(Nα)) ◦Fk(τ′−1
β,α(Ni)) ◦Fk(τ′
β,α(Ni)) ◦F−1
β(D′
α(Ni))◦
Dβ(F−1
α(Ni)) ◦τ−1
β,α(Fi(Ni))
=Fk(Nβ)◦Fk(D′
β(Nα)) ◦F−1
β(D′
α(Ni)) ◦Dβ(F−1
α(Ni)) ◦τ−1
β,α(Fi(Ni))
=Fk(Nβ)◦F−1
β(Nj)◦Dβ(Fj(Nα)) ◦Dβ(F−1
α(Ni)) ◦τ−1
β,α(Fi(Ni))
=F!(N)β◦Dβ(Fj(Nα)) ◦Dβ(F−1
α(Ni)) ◦τ−1
β,α(F!(N)i) by def. of F!(N)β
=F!(N)β◦Dβ(F!(N)α)◦τ−1
β,α(F!(N)i),by def. of F!(N)α
where the seventh equality holds by the commutativity of the diagram induced by applying the
natural isomorphism F−1
βto the morphism Nα. Thus, F!(N) satisfies the axiom (lRep.1).
To verify that F!(N) satisfies the axiom (lRep.2), let ibe an object in Ob(I). Then we have
F!(N)ei◦ηi(F!(N)i) = Fi(Nei)◦F−1
ei(Ni)◦ηi(Fi(Ni)) by def. of F!(N)ei
=Fi(Nei)◦(idFi∗η′
i)(Ni)◦(η−1
i∗idFi)(Ni) by (Mor.2)
=Fi(Nei)◦idFi(D′
ei(Ni)) ◦Fi(η′
i(Ni)) ◦η−1
i(Fi(Ni))◦
idDi(idFi(Ni)) ◦ηi(Fi(Ni)) by def. of ∗
=Fi(Nei)◦Fi(η′
i(Ni))
=Fi(idNi) by (lRep.2)
=idF!(N)i.by def. of F!(N)i
To verify that F!(ω) is a morphism in D-Rep, we have to show that for any morphism α:i→
j∈Mor(I), the diagram
Dα(Fi(Ni)) Dα(Fi(ωi))
//
F!(N)α
Dα(Fi(N′
i))
F!(N′)α
Fj(Nj)Fj(ωj)
//Fj(N′
j)
commutes, that is,
Fj(ωj)◦F!(N)α=F!(N′)α◦Dα(Fi(ωi)).
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 89
Indeed, we have
Fj(ωj)◦F!(N)α=Fj(ωj)◦Fj(Nα)◦F−1
α(Ni) by def. of F!(N)α
=Fj(N′
α)◦Fj(D′
α(ωj)) ◦F−1
α(Ni)
=Fj(N′
α)◦F−1
α(N′
i)◦Dα(Fi(ωi))
=F!(N′)α◦Dα(Fi(ωi)) by def. of F!(N′)α,
where the third equality holds by the commutativity of the diagram induced by applying the natural
isomorphism F−1
αto the morphism ωi.
B.4 Details in the proof of Proposition 3.5. Given two composable morphisms α:i→jand
β:j→kin Mor(I), we get that
U(M′)β◦Dβ(U(M′)α) = M′
β◦F−1
β(M′
j)◦Dβ(M′
α◦F−1
α(M′
i))
=M′
β◦F−1
β(M′
j)◦Dβ(M′
α)◦Dβ(F−1
α(M′
i))
=M′
β◦D′
β(M′
α)◦F−1
β(D′
α(M′
i)) ◦Dβ(F−1
α(M′
i)) by naturality of F−1
β
=M′
βα ◦τ′
β,α(M′
i)◦F−1
β(D′
α(M′
i)) ◦D′
β(F−1
α(M′
i)) by (lRep.1)
=M′
βα ◦τ′
β,α(M′
i)◦(F−1
β∗F−1
α)(M′
i) by def. of F−1
β∗F−1
α
=M′
βα ◦F−1
βα (M′
i)◦τβ,α(M′
i) by (Mor.1)
=U(M′)βα ◦τβ,α(M′
i)
=U(M′)βα ◦τβ,α(U(M′)i).
Thus U(M′) satisfies the axiom (lRep.1).
For i∈Ob(I), we have
idU(M′)i= idM′
i=M′
ei◦η′
i(M′
i) = M′
ei◦F−1
ei(M′
i)◦ηi(M′
i) = U(M′)ei◦ηi(U(M′)i),
where the second equality follows from (lRep.2) and the third equality follows from (Mor.2). Thus
U(M′) satisfies the axiom (lRep.2).
For any morphism α:i→jin Mor(I), we have that
U(K′)α◦Dα(U(δ′)i) = K′
α◦F−1
α(K′
i)◦Dα(δ′
i)
=K′
α◦D′
α(δ′
i)◦F−1
α(M′
i)
=δ′
j◦M′
α◦F−1
α(M′
i) by (†)
=U(δ′)j◦U(M′)α.
Thus U(δ′) is a morphism from U(M′) to U(K′).
90 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
B.5 Details in the proof of Theorem 5.4. For the identity (†), we have:
G!(N)β◦Dβ(G!(N)α)◦τ−1
β,α(G!(N)i)◦χβα ◦sβα,θ
θ
=ϑβ◦χ−1
β◦Dβ(ϑα◦χ−1
α)◦τ−1
β,α(G!(N)i)◦χβα ◦sβα,θ
θby def. of G!(N)βand G!(N)α
=ϑβ◦χ−1
β◦Dβ(ϑα◦χ−1
α)◦τ−1
β,α(G!(N)i)◦Dβα(sθ) by (5.2.2)
=ϑβ◦χ−1
β◦Dβ(ϑα◦χ−1
α)◦Dβ(Dα(sθ)) ◦τ−1
β,α(Dθ(N•))
=ϑβ◦χ−1
β◦Dβ(ϑα)◦Dβ(χ−1
α)◦Dβ(Dα(sθ)) ◦τ−1
β,α(Dθ(N•))
=ϑβ◦χ−1
β◦Dβ(ϑα)◦Dβ(sα,θ
θ)◦τ−1
β,α(Dθ(N•)) by (5.2.2)
=ϑβ◦χ−1
β◦Dβ(sαθ)◦Dβ(τα,θ (N•)) ◦τ−1
β,α(Dθ(N•))
=ϑβ◦χ−1
β◦Dβ(sαθ)◦τ−1
β,αθ (N•)◦τβα,θ(N•) by (Dia.1)
=ϑβ◦sβ,αθ
αθ ◦τ−1
β,αθ (N•)◦τβα,θ (N•) by (5.2.2)
=sβαθ ◦τβα,θ(N•) by (5.2.1)
=ϑβα ◦sβα,θ
θby (5.2.1)
=G!(N)βα ◦χβα ◦sβ α,θ
θ,by def. of G!(N)βα
where the third equality holds by the commutative diagram induced by applying the natural trans-
formation τ−1
β,α to the morphism sθ, and the sixth equality holds by the commutative diagram
induced by applying Dβto (5.2.1). Since χβα is an isomorphism, the universal property of colimits
gives
G!(N)β◦Dβ(G!(N)α)◦τ−1
β,α(G!(N)i) = G!(N)βα.
For the identity (‡), we have:
G!(N)ei◦ηi(G!(N)i)◦sθ=ϑei◦χ−1
ei◦ηi(G!(N)i)◦sθby def. of G!(N)ei
=ϑei◦χ−1
ei◦Dei(sθ)◦ηi(N•)
=ϑei◦sei,θ
θ◦ηi(N•) by (5.2.2)
=sθ◦τei,θ(N•)◦ηi(N•) by (5.2.1)
=sθ,by (Dia.2)
where the second equality holds by the commutative diagram induced by applying the natural
transformation ηito the morphism sθ.
To show the identity (♯), we have
ωj◦G!(N)α◦χα◦sα,θ
θ=ωj◦ϑα◦sα,θ
θby def. of G!(N)α
=ωj◦sαθ ◦τα,θ(N•) by (5.2.1)
=s′
αθ ◦Dαθ(σ•)◦τα,θ (N•) by (5.2.3)
=s′
αθ ◦τα,θ(N′
•)◦Dα(Dθ(σ•))
=ϑ′
α◦s′
α,θ ◦Dα(Dθ(σ•)) by (5.2.1)
=ϑ′
α◦χ′−1
α◦Dα(s′
θ)◦Dα(Dθ(σ•)) by (5.2.2)
=ϑ′
α◦χ′−1
α◦Dα(ωi)◦Dα(sθ)
=ϑ′
α◦χ′−1
α◦Dα(ωi)◦χα◦sα,θ
θby (5.2.1)
=G!(N′)α◦Dα(ωi)◦χα◦sα,θ
θ,by def. of G!(N′)α
REPRESENTATIONS OVER DIAGRAMS OF CATEGORIES AND ABELIAN MODEL STRUCTURES 91
where the fourth equality holds by the commutative diagram induced by applying the natural
transformation τα,θ to the morphism σ•, and the seventh equality holds by the commutative diagram
induced by applying Dαto (5.2.3). Since χαis an isomorphism, the desired equality follows from
the universal property of colimits.
For the identity (♮), we have
G∗(M)λ◦(D◦G)λ(u(ω)i)
=MG(λ)◦DG(λ)(ωG(i))◦DG(λ)(seG(i))◦DG(λ)(ηG(i)(Ni))
=ωG(j)◦G!(N)G(λ)◦DG(λ)(seG(i))◦DG(λ)(ηG(i)(Ni))
=ωG(j)◦ϑG(λ)◦χ−1
G(λ)◦DG(λ)(seG(i))◦DG(λ)(ηG(i)(Ni))
=ωG(j)◦ϑG(λ)◦χ−1
G(λ)◦χG(λ)◦sG(λ),eG(i)
eG(i)◦DG(λ)(ηG(i)(Ni)) by (5.2.2)
=ωG(j)◦ϑG(λ)◦sG(λ),eG(i)
eG(i)◦DG(λ)(ηG(i)(Ni))
=ωG(j)◦sG(λ)◦eG(i)◦τG(λ),eG(i)(Ni)◦DG(λ)(ηG(i)(Ni)) by (5.2.1)
=ωG(j)◦sG(λ)◦eG(i)◦τeG(j),G(λ)(Ni)◦ηG(j)(DG(λ)(Ni)) by (Dia.2)
=ωG(j)◦sG(λ)◦τeG(j),G(λ)(Ni)◦ηG(j)(DG(λ)(Ni))
=ωG(j)◦seG(j)◦DeG(j)(Nλ)◦τ−1
eG(j),G(λ)(Ni)◦τeG(j),G(λ)(Ni)◦ηG(j)(DG(λ)(Ni))
=ωG(j)◦seG(j)◦DeG(j)(Nλ)◦ηG(j)(DG(λ)(Ni))
=ωG(j)◦seG(j)◦DeG(j)(Nλ)◦τ−1
eG(j),G(λ)(Ni) by (Dia.2)
=ωG(j)◦seG(j)◦ηG(j)(Nj)◦η−1
G(j)(Nj)◦DeG(j)(Nλ)◦τ−1
eG(j),G(λ)(Ni)
=ωG(j)◦seG(j)◦ηG(j)(Nj)◦Nλby (lRep.1)
=u(ω)j◦Nλ,
where the first equality holds by the definitions of G∗(M)λ, (D◦G)λand u(ω)i, the third holds by
the definition of G!(N)G(λ), and the ninth holds by the definition of cocone.
For the identity (§), we have
v(σ)j◦G!(N)α◦χα◦sα,θ
θ=v(σ)j◦ϑα◦sα,θ
θ
=v(σ)j◦sαθ ◦τα,θ(N•) by (5.2.1)
=Mαθ ◦Dαθ(σ•)◦τα,θ (N•)
=Mαθ ◦τα,θ(G∗(M)•)◦Dα(Dθ(σ•))
=Mα◦Dα(Mθ)◦Dα(Dθ(σ•)) by (lRep.1)
=Mα◦Dα(v(σ)i)◦Dα(sθ)
=Mα◦Dα(v(σ)i)◦χα◦sα,θ
θ,by (5.2.2)
where the first equality holds by the definition of G!(N)α, the third holds by (5.4.1), the fourth
holds by the commutative diagram induced by applying the natural transformation τα,θ to the
morphism σ•, and the sixth holds by the commutative diagram induced by applying Dαto (5.4.1).
Acknowledgments
We thank Zhongkui Liu and Fei Xu for helpful discussions related to this work. Z. Di was partly
supported by NSF of China Grant No. 11971388; L. Li was partly supported by NSF of China
Grant No. 12171146 and the Hunan Provincial Science and Technology Department Grant No.
2019RS1039; L. Liang was partly supported by NSF of China Grant No. 12271230 and NSF of
92 Z.X. DI, L.P. LI, L. LIANG, AND N.N. YU
Gansu Province Grant No. 21JR7RA297; N. Yu was partly supported by NSF of China Grant No.
11971396.
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Z.X. Di School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
Email address:dizhenxing@163.com
L.P. Li LCSM (Ministry of Education), Department of Mathematics, Hunan Normal University,
Changsha 410081, China.
Email address:lipingli@hunnu.edu.cn
L. Liang Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
Email address:lliangnju@gmail.com
URL:https://sites.google.com/site/lliangnju
N.N. Yu School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Email address:ninayu@xmu.edu.cn
