Higher Segal structures in algebraic $K$-theory

Abstract

We introduce higher dimensional analogues of simplicial constructions due to Segal and Waldhausen, respectively producing the direct sum and algebraic $K$-theory spectra of an exact category. We then investigate their fibrancy properties, based on the formalism of higher Segal spaces by Dyckerhoff-Kapranov.

Full text

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY
THOMAS POGUNTKE
Abstract. We introduce higher dimensional analogues of simplicial constructions due
to Segal and Waldhausen, respectively producing the direct sum and algebraic K-theory
spectra of an exact category. We interrelate them by totalizing, and investigate their
fibrancy properties based on the formalism of higher Segal spaces of Dyckerhoff-Kapranov.
Contents
Introduction 1
1. Cyclic polytopes 3
2. Higher Segal conditions 5
3. The higher Segal construction 10
4. The higher Waldhausen construction 15
5. Stringent categories and path spaces 23
References 35
Introduction
The notion of a Segal object was introduced by Rezk, motivated by the observation that
a simplicial set satisfies the Segal conditions if and only if it is the nerve of a category. In
this sense (and much more generally), they model the structure of an associative monoid.
The monoid structure comes from the correspondence
X1×X0X1
(∂2,∂0)
←−−−− X2
∂1
−−−−→ X1,(0.1)
which is a bona fide map if Xis Segal. The definition of higher Segal objects is motivated by
the question when (0.1) still satisfies the associativity constraints as a multi-valued operation.
The answer is provided by the 2-Segal conditions, which were introduced in [11].
Namely, Xis 2-Segal if Xnis recovered as a certain limit over its 2-skeleton in two different
ways, described by the combinatorics of triangulations of the cyclic polygon on n+ 1 vertices.
As suggested in op.cit., this allows for a natural extension to any dimension d≥0, based
on the theory of d-dimensional cyclic polytopes and their triangulations. These higher Segal
structures are introduced in this work. A categorical reason for cyclic polytopes to appear is
that they provide a model for the free ω-categories generated by simplices.
Each cyclic polytope has two distinguished triangulations, obtained as projections of the
lower and upper boundaries in the next higher dimension, which allow for a purely combina-
torial description. The lower (resp. upper) d-Segal condition requires that Xnbe recovered
as the corresponding limit over its d-skeleton, for all n≥d.
From a different perspective, unital 2-Segal spaces were defined and studied independently
in [14] and its series of sequels under the name decomposition space. Further work in this
area includes a precise description of unital 2-Segal sets in terms of double categories in [2],
and the introduction of relative Segal conditions in [27] and [30], which model the structure
Date: August 19, 2018.
2010 Mathematics Subject Classification. 18E05, 18E10, 18G30, 19D10, 55U10.
Key words and phrases. Algebraic K-theory, higher Segal spaces, Waldhausen S-construction.
1

2 THOMAS POGUNTKE
of modules over multi-valued monoids. Moreover, in [28], unital 2-Segal spaces are shown to
be equivalent to certain ∞-operads.
The first main objective of this work is to investigate the first examples of d-Segal objects,
where d > 2, explicitly appearing in the literature. These simplicial categories naturally
arise in the algebraic K-theory of exact categories E. To wit, they are higher dimensional
generalizations of Segal’s construction S⊕(E) in [25] and Waldhausen’s construction S(E)
in [29], which provide deloopings of the direct sum and algebraic K-theory spectra of E,
respectively.
For k= 2, the k-dimensional Segal and Waldhausen constructions Shki
⊕(E), resp. Shki(E),
first appeared in the context of real algebraic K-theory as introduced by Hesselholt-Madsen
[18], allowing the construction of genuine Gal(C|R)-equivariant deloopings in the presence of
a duality structure on E, which was further studied for example by Dotto [6].
The extension to arbitrary k≥1 is straight-forward; instead of a type of configuration
space on the circle (in Segal’s context) or the 2-sphere (Hesselholt-Madsen), we consider k-
dimensional spheres. On the other hand, diagrams of short exact sequences (Waldhausen)
and 2-extensions (Hesselholt-Madsen) are replaced by (k+ 2)-term exact sequences.
Beyond being an evident generalization, Shki(E) also appears naturally on the simplicial
side of the secondary Dold-Kan correspondence [10] in the case where Eis a stable ∞-category,
as a categorified Eilenberg-MacLane space.
It was already shown by Segal that Sh1i
⊕(E) satisfies the (lower 1-)Segal conditions, which
generalize as follows.
Theorem 0.1. The higher Segal construction Shki
⊕(E)is a lower (2k−1)-Segal category.
Similarly, for the Waldhausen construction, the 2-Segal property of Sh1i(E) is one of the
main results of [11]. Our first generalization of it puts some homological algebraic constraints
on E, as illustrated by the following statement.
Theorem 0.2. The simplicial category Sh2i(E)is upper 3-Segal if and only if Eis an abelian
category.
Interestingly, this case is an outlier – in general, we only have the following slightly weaker
result (any d-Segal property implies all higher dimensional ones).
Theorem 0.3. The higher Waldhausen construction Shki(E)of the exact category Eis 2k-
Segal.
The main step in the proof for abelian Eis the calculation of the path spaces of Shki(E)
in order to make use of another inductive property of the Segal conditions; this also works
for weakly idempotent complete E. On the lowest simplicial level, this amounts to a version
of the Noether isomorphism theorem for longer left exact sequences (or a higher octahedral
axiom if Eis derived; interestingly, this also applies to the theory of [19]).
From an algebraic perspective, our results lay the groundwork for further investigation
of the algebraic structures induced by the Segal and Waldhausen constructions (cf. below).
From a topological perspective, they are interesting since they provide new deloopings of the
direct sum and algebraic K-theory of E. Historically seen, much progress has been made due
to the existence of new models which exhibit specific features more clearly than others, e.g.
power operations in algebraic K-theory due to Grayson [15] and Nenashev [21].
Theorem 0.4. There is a natural equivalence of simplicial categories from the k-dimensional
Waldhausen construction of Eto the total simplicial object of its k-fold S-construction,
Shki(E)'
−−→ T S(k)(E).
Similarly, there is an equivalence of simplicial categories
Shki
⊕(E)'
−−→ T S(k)
⊕(E).
In particular, this induces equivalences on geometric realizations, from which we deduce
the appropriate delooping and additivity theorems.

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 3
Corollary 0.5. (Delooping). The algebraic K-theory spectrum of Eis equivalent to
|Sh0i(E)'| −→ Ω|Sh1i(E)'| −→ Ω2|Sh2i(E)'| −→ . . .
where (−)'denotes the maximal subgroupoid of a category.
(Additivity). The functor (∂2, ∂0)on S2(E)induces a homotopy equivalence
|Shki(S2(E))'|'
−−→ |Shki(E)'|×|Shki(E)'|,
More precisely, the simplicial space |Shki(S•(E))'|is a Segal space.
Let us briefly outline the structure of the paper. In §1, we summarize some basic theory
of cyclic polytopes and their triangulations from [23] and [31], which is used in §2 to define
and study relations between the higher Segal conditions from [12]. Novel results concern the
interaction with diagonals and total simplicial objects of multisimplicial objects.
In §3, we define the simplicial category Shki
⊕(E), realize it as the total simplicial object of the
iterated 1-dimensional construction, and prove Theorem 0.1. Section 4 provides definitions
of the higher Waldhausen construction and first examples; we show that Shki(E) is the total
simplicial object of Waldhausen’s original construction, from which we deduce delooping and
additivity theorems, as well as Theorem 0.2.
Finally, §5 introduces the requisite homological context enabling us to establish further
Segal properties for the higher S-constructions; it also includes some further examples.
Acknowledgements
I would like to thank T. Dyckerhoff for suggesting this topic, his constant interest and
invaluable contributions, as well as several enlightening discussions with G. Jasso. I would
also like to thank W. Stern for helpful suggestions, as well as M. Penney and T. Walde for
fruitful discussions. I thank A. Cegarra for pointing out the proof of Lemma 3.9 to me.
1. Cyclic polytopes
In this section, we recall results on polytopes relevant to the study of higher Segal objects.
Definition 1.1. Let d≥0, and consider the moment curve
γd:R−→ Rd, t 7−→ (t, t2, t3, . . . , td).
For a finite subset N⊆R, the d-dimensional cyclic polytope on the vertices Nis defined to
be the convex hull of the set γd(N)⊆Rd, and denoted by
C(N, d) = conv(γd(N)).
The combinatorial type of the polytope C(N, d) only depends on the cardinality of N. We
will usually consider Nto be the set [n] = {0,1, . . . , n}, where n≥0.
Cyclic polytopes are simplicial polytopes, i.e., their boundary forms a simplicial complex,
which organizes into two components (with non-empty intersection), as follows.
Definition 1.2. A point xin the boundary of C([n], d + 1) is called a lower point, if
(x−R>0)∩C([n], d + 1) = ∅,
where the half-line of positive real numbers R>0⊆Rd+1 is embedded into the last coordinate.
Similarly, xis said to be an upper point, if
(x+R>0)∩C([n], d + 1) = ∅.
The lower and upper points of the boundary form simplicial subcomplexes of C([n], d + 1),
which admit the following purely combinatorial description.
Definition 1.3. Let I⊆[n]. A gap of Iis a vertex j∈[n]rIin the complement of I. A
gap is said to be even, resp. odd, if the cardinality #{i∈I|i > j}is even, resp. odd. The
subset Iis called even, resp. odd, if all gaps of Iare even, resp. odd.

4 THOMAS POGUNTKE
Proposition 1.4 (Gale’s evenness criterion; [31], Theorem 0.7).Let n≥0, and let I⊆[n]
with #I=d+1. Then Idefines a d-simplex in the lower, resp. upper, boundary of C([n], d+1)
if and only if Iis even, resp. odd.
Forgetting the last coordinate of Rd+1 defines a pro jection map
p:C([n], d + 1) −→ C([n], d).
For any I⊆[n] with #I−1 = r≤d, the projection pmaps the geometric r-simplex
|∆I| ⊆ C([n], d + 1)
homeomorphically to an r-simplex in C([n], d).
Definition 1.5. The lower triangulation Tof the polytope C([n], d) is the triangulation given
by the projections under pof the simplices contained in the lower boundary of C([n], d + 1).
Similarly, the upper triangulation Tuof C([n], d) is defined by the projections of the simplices
contained in the upper boundary of the polytope C([n], d + 1).
Vice versa, any triangulation of C([n], d) induces a piecewise linear section of p, whose
image defines a simplicial subcomplex of C([n], d + 1). This interplay between the cyclic
polytopes in different dimensions is what makes their combinatorics comparatively tractible.
Definition 1.6. Given a set I ⊆ 2[n]of subsets of [n], we denote by
∆I⊆∆n(1.1)
the simplicial subset of ∆nwhose m-simplices are given by those maps [m]→[n] which factor
over some I∈ I.
From the above discussion, it follows that we have canonical homeomorphisms
|∆T|∼
−−→ C([n], d),and |∆Tu|∼
−−→ C([n], d),
expressing the lower, resp. upper, triangulation of C([n], d) geometrically.
Definition 1.7. Let I, J ⊆[n] be subsets of cardinality d+ 1, as well as |∆I|and |∆J|the
geometric d-simplices in C([n], d)⊆Rdthey define, respectively. Let L(|∆J|) denote the set
of lower boundary points of |∆J|=C(J, d), and similarly, let U(|∆I|) be the set of upper
boundary points of |∆I|=C(I , d). We write
|∆I|≺|∆J| ⇐⇒ |∆I|∩|∆J| ⊆ U(|∆I|)∩L(|∆J|).
If |∆I|≺|∆J|, then we say that |∆I|lies below the simplex |∆J|.
Proposition 1.8 ([23], Corollary 5.9).The transitive closure of ≺defines a partial order on
the set of nondegenerate d-simplices in ∆n.
Remark 1.9. Suppose that T ⊆ 2[n]consists of subsets of [n] of cardinality d+ 1 and defines
a triangulation of the cyclic polytope C([n], d). In particular,
|∆T|∼
=C([n], d).
Let I0∈ T . Then either L(|∆I0|) is contained in L(|∆T|), or there is some I1∈ T such that
the simplex |∆I0|lies below |∆I1|. Iterating this argument, we obtain a chain
|∆I0|≺|∆I1| ≺ |∆I2| ≺ . . .
of subsimplices of |∆T|. The statement of Proposition 1.8 implies that this chain is acyclic
and therefore has to terminate after finitely many steps. Thus, there exists I∞∈ T with
L(|∆I∞|)⊆L(|∆T|).
Similarly, there exists a set I−∞ ∈ T such that U(|∆I−∞|)⊆U(|∆T|).

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 5
2. Higher Segal conditions
Let Cbe an ∞-category which admits finite limits. Following [12], we introduce a frame-
work of fibrancy properties of simplicial objects in Cgoverned by the combinatorics from §1
of cyclic polytopes and their triangulations.
Definition 2.1. For n≥d≥0, we introduce the lower subposet of 2[n]as follows;
L([n], d) = {J|J⊆Ifor some even I⊆[n] with #I=d+ 1}.
Analogously, we define
U([n], d)⊆2[n]
as the poset of all subsets contained in an odd subset I⊆[n] of cardinality #I=d+ 1.
By Proposition 1.4, the sets of subsimplices of C([n], d) described by L([n], d) and U([n], d)
define the lower and upper triangulations of the cyclic polytope, respectively.
Definition 2.2. Let d≥0. A simplicial object X∈ C∆is called
•lower d-Segal if, for every n≥d, the natural map
Xn−→ lim
←−
I∈L([n],d)
XI
is an equivalence;
•upper d-Segal if, for every n≥d, the natural map
Xn−→ lim
←−
I∈U([n],d)
XI
is an equivalence;
•d-Segal if Xis both lower and upper d-Segal.
Example 2.3. Let X∈ C∆be a simplicial object.
(1) Then Xis lower (or upper) 0-Segal if and only if X'
−−→ X0is equivalent to the
constant simplicial object on its 0-cells.
(2) The simplicial object Xis lower 1-Segal if, for every n≥1, the map
Xn−→ X{0,1}×X{1}X{1,2}×X{2}· · · ×X{n−1}X{n−1,n}
is an equivalence. That is to say, Xis a Segal object in the sense of Rezk [24].
For X∈Set∆, this means that Xis equivalent to the nerve of the category with
objects X0, morphisms X1, and composition induced by the correspondence
X{0,1}×X{1}X{1,2}∼
←−− X2−→ X{0,2}.
Furthermore, X∈Set∆is 1-Segal if and only if this defines a discrete groupoid. In
fact, an upper 1-Segal set is a quiver; in general, an object X∈ C∆is upper 1-Segal
if, for every n≥1, we have
Xn'
−−→ X{0,n}.
(3) The simplicial object Xis lower 2-Segal if, for every n≥2, the map
Xn−→ X{0,n−1,n}×X{0,n−1}X{0,n−2,n−1}×X{0,n−2}· · · ×X{0,2}X{0,1,2}
is an equivalence. Similarly, Xis upper 2-Segal if, for every n≥2, we have
Xn'
−−→ X{0,1,n}×X{1,n}X{1,2,n}×X{2,n}· · · ×X{n−2,n}X{n−2,n−1,n}.
It follows that Xis 2-Segal if and only if it is 2-Segal in the sense of [11]. This is
most readily seen by reducing to Segal objects as in (2) by applying the respective
path space criteria, Proposition 2.8 and [11], Theorem 6.3.2 – or by Proposition 2.6.
Lemma 2.4. Let d≥0. A limit of lower, resp. upper, d-Segal objects is again lower, resp.
upper, d-Segal.
Proof. This is clear, because all the conditions are defined in terms of finite limits. 

6 THOMAS POGUNTKE
Remark 2.5. Let Xbe a simplicial object in an ∞-category Cwhich admits limits. Then
we can form the right Kan extension of Xalong the (opposite of the) Yoneda embedding
Yop :N(∆op)−→ Fun(N(∆op),Top)op,
where Top denotes the ∞-category of spaces. In particular, by means of this extension, we
may evaluate Xon any simplicial set. Then we can reformulate the higher Segal conditions
as follows. Let Σd= Σ|d∪Σu|ddenote the collection of d-Segal coverings, where
Σ|d={∆L([n],d)→∆n|n≥d},and Σu|d={∆U([n],d)→∆n|n≥d},
see [11], §5.2. Then Xis d-Segal if and only if it is Σd-local (cf. loc.cit., Proposition 5.1.4).
In this case, we say that the elements of Σdare X-equivalences (meaning, Xmaps them to
equivalences in C).
Similarly, Xis lower, resp. upper, d-Segal if and only if it is Σ|d-local, resp. Σu|d-local.
Proposition 2.6. Let d≥0, and let Xbe a simplicial object in an ∞-category Cwith limits.
Then Xis d-Segal if and only if, for every n≥d, and every triangulation of the cyclic
polytope C([n], d)defined by the poset of simplices T ⊆ 2[n], the natural map
Xn−→ lim
←−
I∈T
XI
is an equivalence.
Proof. By [23], Corollary 5.12, any triangulation Tof C([n], d) can be connected to the lower
and upper triangulations Tand Tuvia sequences of elementary flips of the form
L(|∆I|)⊆ |∆I| ⊇ U(|∆I|).
This implies that we have a zig-zag of X-equivalences of the form
∆L([n],d)−→ . . . ←− ∆T−→ . . . ←− ∆U([n],d)
in the category of simplicial sets over ∆n. This implies that the inclusion ∆T⊆∆nis again
an X-equivalence by 2-out-of-3, which was to be shown.
For the converse, there is nothing to prove. 
Definition 2.7. Let Xbe a simplicial object in an ∞-category C. The left path space PX
is the simplicial object in Cdefined as the pullback of Xalong the endofunctor
c: ∆ →∆,[n]7→ [0] ⊕[n].
Here, for two linearly ordered sets Iand J, the ordinal sum I⊕Jis the disjoint union IqJ
of sets, endowed with the linear order where i≤jfor every pair of i∈Iand j∈J.
Similarly, the right path space PXis given by the pullback of Xalong
c: ∆ →∆,[n]7→ [n]⊕[0].
Proposition 2.8 (Path space criterion).Let d≥0, and let Cbe an ∞-category with finite
limits. Let X∈ C∆be a simplicial object.
•Suppose dis even. Then
(1) Xis lower d-Segal if and only if PXis lower (d−1)-Segal,
(2) Xis upper d-Segal if and only if PXis lower (d−1)-Segal.
•Suppose dis odd. Then the following conditions are equivalent.
(i)Xis upper d-Segal.
(ii)PXis upper (d−1)-Segal.
(iii)PXis lower (d−1)-Segal.
Proof. We show that if dis even, then Xis an upper d-Segal object if and only if PXis
lower (d−1)-Segal. All other assertions follow by analogous arguments.
Let ([n], d) denote the set of maximal elements of the poset L([n], d), so that
∆L([n],d)=[
I∈([n],d)
∆I

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 7
is the minimal presentation, and similarly for u([n], d)⊆ U([n], d).
Then the claim is an immediate consequence of the following observation. A subset I⊆[n]
of cardinality d+ 1 is odd if and only if n∈Iand Ir{n}is an even subset of [n−1]. Thus
the map c:([n−1], d −1) →u([n], d), I7→ I⊕[0], is a bijection. 
Remark 2.9. There is no path space criterion for lower d-Segal objects if dis odd. While
([5],3) = {{0,1,2,3},{0,1,3,4},{0,1,4,5},{1,2,4,5},{2,3,4,5},{1,2,3,4}},
the maps c:([4],2) →([5],3), I7→ [0] ⊕I, and c:u([4],2) →([5],3), I7→ I⊕[0], are
not even jointly surjective, and their images
im(c) = {{0,1,2,3},{0,1,3,4},{0,1,4,5}},im(c) = {{0,1,4,5},{1,2,4,5},{2,3,4,5}}
intersect. Rather, the complement of im(c) in ([n], d) is given by ([n−1], d), for all d∈N
and n≥d. By induction, we obtain a disjoint decomposition of sets
([n], d) =
n
a
j=d
u([j−1], d −1) ⊕ {j}.(2.1)
In fact, this statement, resp. its dual, also follow by grouping the elements of ([n], d) by
their maximal, resp. minimal, vertex, and the following explicit description. If dis odd, then
([n], d) = na
i∈J
{i, i + 1}J⊆[n] with #J=d+ 1
2and |i−j|>1 for all i, j ∈Io,(2.2)
for all n≥d. All other ([n], d) and u([n], d) are described from this by applying the proof
of Proposition 2.8.
Proposition 2.10. Let Xbe a simplicial object in an ∞-category Cwhich admits limits.
Assume that Xis lower or upper d-Segal. Then Xis (d+ 1)-Segal.
Proof. We show the statement assuming that Xis lower d-Segal. The proof for upper d-Segal
objects is similar. Let n≥d+ 1 and consider a collection Tdefining a triangulation |∆T|
of the cyclic polytope C([n], d + 1). Recall that Tdefines the simplicial complex L(|∆T|) of
lower facets, and the projection p:C([n], d + 1) →C([n], d) identifies |∆T| ⊆ C([n], d + 1)
with the simplicial subcomplex defining the lower triangulation
p(|∆T|)⊆C([n], d).
Thus, we obtain a commutative diagram
∆T∆n
∆T
ι
κι
(2.3)
of simplicial sets, in which by definition, ι∈Σd+1 and ι∈Σ|d. In order to deduce (d+ 1)-
Segal descent for X, we have to show that ι∈Σ|d, the collection of Σ|d-equivalences, by
the tautological part of Proposition 2.6.
By the 2-out-of-3 property of Σ|d, it suffices to show that κ∈Σ|d. We will do so by
showing that κcan be obtained as an iterated pushout along morphisms in Σ|d.
By Remark 1.9, the triangulation |∆T|of C([n], d + 1) contains a maximal (d+ 1)-simplex
of the form |∆I|, defined by a singleton collection {I} ⊆ [n]
d+2. Let Ibe the set which
defines the lower facets of |∆I|, defining a triangulation |∆I|of C(I, d). Then the inclusion
of simplicial sets
κ: ∆I−→ ∆I

8 THOMAS POGUNTKE
is contained in Σ|d. Further, since |∆I|is a maximal simplex, we have a pushout diagram
∆T∆I
∆T(1) ∆I
·y
κ(1) κ
of simplicial sets, where T(1) =Tr{I}. Thus, the map κ(1) lies in Σ|d, and |∆T(1) |is an
admissible simplicial subcomplex of C([n], d + 1) with one (d+ 1)-simplex less than |∆T|.
Assume that the triangulation |∆T|consists of exactly rsimplices of dimension (d+ 1).
By iterating the argument just given, we obtain a chain of morphisms
∆T−→ ∆T(r−1) −→ . . . −→ ∆T(1) −→ ∆T
in Σ|dwhose composite is the morphism κfrom (2.3). Thus, it is also contained in Σ|d.
Definition 2.11. Let Cbe an ∞-category with finite limits, and Y∈ C∆×kak-fold simplicial
object, for k≥0. We will denote by DY ∈ C∆the diagonal of Y, which is defined to be the
pullback of Yunder the diagonal embedding diag: ∆ →∆×k.
Remark 2.12. Let Cbe an ∞-category with finite limits, and let Y∈ C∆×kbe a k-fold
simplicial object, and let r≥0. We conjecture the following compatibilities.
(1) If Yis k-fold lower (2r−1)-Segal, r≥1, then DY is lower (2kr −1)-Segal.
(2) If Yis k-fold lower, resp. upper, 2r-Segal, then DY is upper, resp. lower, 2kr-Segal.
(3) If Yis k-fold upper (2r+ 1)-Segal, then DY is upper (2kr + 1)-Segal.
Note that (1) implies the other two statements, by repeated applications of the path space
criterion; indeed, PDY 'DP 
(k)Yand PDY 'DP 
(k)Y, where the notation P
(k)and P
(k)
means application of the corresponding path space functor in each variable.
Definition 2.13. Let V:C∆→ C∆×kbe the total d´ecalage, which is the pullback along the
k-fold ordinal sum ⊕: ∆×k→∆. The total simplicial object of Y∈ C∆×kis denoted by
T Y ∈ C∆,
where Tis defined as the right adjoint of V.
Remark 2.14. Let Y∈ C∆×kbe a k-fold simplicial object. If k= 2, then for all n≥0, by
a cofinality argument, the n-cells of the total simplicial object of Yare computed by
(T Y )n'
−−→ eq n
Q
i=0
Y{0,...,i},{i,...,n}Q
I⊕J=[n]
YI,J ,
ψ
χ
where the components of ψare given by ∂•,0◦prifor I={0, . . . , i}, and χconsists of the
functors ∂i,•◦prifor J={i, . . . , n}, cf. [1], §III. Equivalently, this can be expressed as
(T Y )n'
−−→ lim
←−
I1∪...∪Ik=[n]
I1≤...≤Ik
YI1,...,Ik(2.4)
for arbitrary k≥1, by induction.
Definition 2.15. Let d= (di)∈Nk, and let Y∈ C∆×kbe a k-fold simplicial object. We
say that Yis (lower, resp. upper) d-Segal if it is (lower, resp. upper) di-Segal in the ith
simplicial direction, for all 1 ≤i≤k.
Proposition 2.16. Let Cbe an ∞-category with finite limits, r∈Nk, and let Y∈ C∆×kbe
a lower (2r−1)-Segal object. Then T Y ∈ C∆is lower (2r−1)-Segal, where r=Pri.
Proof. For k= 1, this is a tautology. Now assume the statement for some k≥1, and consider
a lower (2r−1)-Segal object Y=Y•,•∈(C∆)∆×kin C∆which is lower (2s−1)-Segal in the

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 9
remaining coordinate. By induction, T Ym,•is lower (2r−1)-Segal, for all m≥0, and we can
reduce to the case k= 2. Thus, we want to prove that, for all n≥2(r+s)−1, the inclusion
[
H∈[
H=I∨J
I≤J
∆I×∆J−→
n
[
i=0
∆{0,...,i}×∆{i,...,n}(2.5)
is a Y-equivalence, where =([n],2(r+s)−1), and I∨Jsignifies the non-disjoint union.
By eliminating redundant summands, we can write the left-hand side as
[
H∈[
H=I∨J, I≤J
2r≤#I≤2r+1
∆I×∆J∪
2r−1
[
i=0
∆[i]×∆Hr[i−1] ∪
n−1
[
j=n−2s
∆H∩[j+1] ×∆[n]r[j].
Combining this, as well as the lower (2s−1)-Segal property in the other variable, with
Proposition 2.10, and using (2.2), the statement follows. 
Remark 2.17. Let C= Set. Then the functor defined by Proposition 2.16,
T:{k-fold categories} −→ {lower (2k−1)-Segal sets}(2.6)
is not an equivalence. In fact, for k= 2, the main result of [2] shows that the functor
V:{2-Segal sets} −→ {double categories}
induces an equivalence between the subcategory of unital 2-Segal sets on the left- and stable
double categories together with the extra datum of an augmentation on the right-hand side.
In particular, this implies that (2.6) cannot be fully faithful.
It is an interesting problem to generalize the above equivalence to arbitrary k. The defini-
tion of stability directly extends to k-fold categories (in terms of (k+ 1)-cubes), and so does
the notion of augmentation; the analogue of unitality for higher Segal objects should involve
degenerate triangulations of the appropriate cyclic polytopes.
Our considerations in §4 suggest a candidate for the inverse functor; on the other hand, it
appears that for Vto produce k-fold Segal objects, it still needs to be restricted to 2-Segal
objects.
Our next result is the analogue of Remark 2.12 (2) for the total simplicial object; however,
the proof is not as straight-forward, for the following reason.
Lemma 2.18. Let Cbe an ∞-category with finite limits, and let Y∈ C∆×kbe a k-fold
simplicial object, k≥1. Then PT Y 'T P Yand PT Y 'T P Y, where P, resp. P, is
applied in the first, resp. last, simplicial direction.
Proof. This is an immediate consequence of the following base change squares,
∆×k∆×k∆×k∆×k
∆ ∆ ∆ ∆
⊕
c×id
⊕ ⊕
id ×c
⊕
cc
where cand care the maps from Definition 2.7. 
Proposition 2.19. Let Cbe an ∞-category with finite limits. Let k≥1, and let Y∈ C∆×k
be a 2r-Segal object, r∈Nk. Then T Y ∈ C∆is a 2r-Segal object, where r=Pri.
Proof. For k= 1, there is nothing to show. In order to verify the upper 2k-Segal condition
by induction, we use the path space criterion as well as (2.1) to obtain
u([n],2k) = ([n−1],2k−1) ⊕ {n}=
n−1
a
j=2k−1
u([j−1],2k−2) ⊕ {j, n}.(2.7)

10 THOMAS POGUNTKE
This suffices, since for the lower 2k-Segal condition, we can use the dual decomposition,
([n],2k) =
n−2k+1
a
i=1
{0, i} ⊕ ([n]r[i],2k−2).
Now let k= 2. We aim to exhibit the n-cells of the total simplicial object as the limit
(T Y )n'
−−→ lim
←−
H∈U([n],4)
lim
←−
I∪J=H
I≤J
YI,J .
By (2.7), the H∈u([n],4) are precisely H={i−1, i, j −1, j, n}with 0 < i < j −1< n −1.
The corresponding factor in the limit is of the following form,
Y(i−1)i,i(j−1)jn ×Y(i−1)i,(j−1)jn Y(i−1)i(j−1),(j−1)j n ×Y(i−1)i(j−1),jn Y(i−1)i(j−1)j,jn
'Y(i−1)i,i(j−1)n×Y(i−1)i,(j−1)nY(i−1)i(j−1),(j−1)jn ×Y(i−1)(j−1),jn Y(i−1)(j−1)j,j n
(2.8)
by the upper 2-Segal property in the last and the lower 2-Segal property in the first coordinate,
respectively. But the factors of the form Y(i−1)i,i(j−1)nand Y(i−1)(j−1)j,jn cancel precisely
with the non-maximal elements of U([n],4), with the exception of the extremal cases Y01,12n
and Y0(n−2)(n−1),(n−1)n, respectively.
On the other hand, we can describe the n-cells of the total simplicial object as follows,
(T Y )n'Y01,1...n ×Y01,2...n Y012,2...n ×Y012,3...n . . . ×Y0...(n−3),(n−2)(n−1)n
Y0...(n−2),(n−2)(n−1)n×Y0...(n−2),(n−1)nY0...(n−1),(n−1)n
'Y01,12n×Y01,2nY012,23n×Y012,3n. . . ×Y0...(n−3),(n−2)(n−1)n
Y0...(n−2),(n−2)(n−1)n×Y0(n−2),(n−1)nY0(n−2)(n−1),(n−1)n
(2.9)
by applying the upper 2-Segal property in the first variable wherever possible, as well as the
lower 2-Segal property in the last variable to the final factor. But the factor Y0...(j−1),(j−1)jn
is precisely the limit of the (2.8) after cancellation for all 0 < i < j −1. 
Definition 2.20. The edgewise subdivision functor E:C∆→ C∆is the pullback along
e: ∆ −→ ∆,[n]7−→ [n]op ⊕[n].
Remark 2.21. By the work [3] of Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer, a
simplicial object X∈ C∆is 2-Segal if and only if EX ∈ C∆is Segal. Again, it would be very
interesting to find an analogous result for the higher Segal conditions.
3. The higher Segal construction
Let Dbe a pointed category with finite products. In this section, we study a generalization
of a construction due to Segal [25] which is similar to a construction proposed (for k= 2) by
Hesselholt and Madsen [18]. The higher dimensional variants (for k≥3) are straightforward
to define, but do not seem to have appeared in the literature as of yet.
Let Fin×denote the category of finite pointed sets. For T∈Fin×, we denote by P(T) its
poset of pointed subsets, considered as a small category.
Definition 3.1. Let T∈Fin×be a finite pointed set. A D-valued presheaf
F:P(T)op −→ D
on P(T) is called a sheaf if, for every pointed subset U⊆T, the canonical map
F(U)−→ Y
u∈Ur{∗}
F({∗, u})
is an isomorphism. We denote by Sh(T , D) the category of D-valued sheaves on P(T).

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 11
Given a map ρ:T→T0in Fin×, we define the pointed preimage functor
ρ×:P(T0)−→ P(T), U 7−→ ρ−1(Ur{∗})q {∗}.
Then the direct image functor F 7→ ρ∗F=F ◦ ρ×makes the assignment
Sh(−,D): Fin×−→ Cat
into a functor with values in the category of small categories.
Definition 3.2. Let k≥1. The k-dimensional Segal construction of Dis defined to be the
simplicial category
Shki
⊕(D) = Sh(Sk,D)∈Cat∆,
where Sk= ∆k/∂∆kis considered as a simplicial object in Fin×.
Lemma 3.3. Let 0≤i≤n. The face map ∂i: Sh(Sk
n,D)→Sh(Sk
n−1,D)is an isofibration.
Proof. Let ρi:Sk
n→Sk
Ibe the corresponding map in Fin×, where I= [n]r{i}. Given
Φ: (ρi)∗F=∂i(F)∼
−−→ G0,
we extend G(ρ×
iV) := G0(V) for V∈ P(Sk
I) by
G(U) := Y
α∈U
α|I6≡∗
F({∗, α})
otherwise. This is functorial, since Dis pointed, and so Φ exhibits G(ρ×
iV) as the product
G(ρ×
iV) = Y
α|I∈Vr{∗}
F({∗, α}).
The lifting of Φ itself is then tautological. 
The goal of this section is to prove the following result, which is due to Segal [25], §2,
for k= 1. Throughout, a lower, resp. upper, d-Segal category means a lower, resp. upper,
d-Segal object in Cat, which is not to be confused with a Segal category in the sense of [8].
Theorem 3.4. Let k≥1, and Da pointed category with finite products. The k-dimensional
Segal construction Shki
⊕(D)is a lower (2k−1)-Segal category. In particular, it is 2k-Segal.
We begin with a proof by direct calculation; a somewhat more conceptual argument will
be provided by Theorem 3.6.
Proof. The last part is an application of Proposition 2.10. Now let n≥2k−1, and set
L=L([n],2k−1).
For I , J ∈ L with I⊇J, we denote by ρI,J :Sk
I→Sk
Jthe corresponding map of pointed sets,
and further write ρI=ρ[n],I for brevity. We have to show that the canonical functor
Sh(Sk
n,D)−→ lim
←−
I∈L
Sh(Sk
I,D) (3.1)
is an equivalence of categories. Note that by Lemma 3.3, all transition maps on the right-hand
side are isofibrations, so that the limit is 1-categorical. Now consider the functor
P:L −→ Cat, I 7−→ P(Sk
I).
We form the following version of its Grothendieck construction
π:XP−→ Lop.
The category XPhas objects (I , U), where I∈ L and U∈ P(Sk
I), and there is a unique
morphism (I, U)≤(J, V ) if I⊇Jand U⊆ρ×
I,J V. The functor πis a cartesian fibration,
where a morphism (I, U)≤(J, V ) is cartesian if
U=ρ×
I,J V.

12 THOMAS POGUNTKE
The category lim
←−I∈L Sh(Sk
I,D) can be identified with the full subcategory of Fun(Xop
P,D)
spanned by those presheaves Fwhich satisfy the following conditions.
(a) The presheaf Fmaps cartesian morphisms to isomorphisms in D.
(b) For every I∈ L, the restriction of Fto the fibre π−1(I) = P(Sk
I) is a sheaf.
AD-valued presheaf on P(Sk
n) defines a presheaf on XPvia pullback along the functor
ϕ0:XP−→ P(Sk
n),(I, U)7−→ ρ×
IU.
The lower (2k−1)-Segal functor (3.1) is then obtained by restricting this pullback along ϕ0
to the category of sheaves on P(Sk
n).
Since the functor ϕ0maps cartesian morphisms to the identity map in P(Sk
n), it factors
over a unique functor ϕ:LXP→ P(Sk
n), where LXPdenotes the localization of XPalong
the set of cartesian morphisms. Note further that imposing condition (a) on a presheaf Fon
the category XPis equivalent to the requirement that Ffactors through LXP.
We obtain an adjunction of presheaf categories as follows,
ϕ!:DLXP←→ DP(Sk
n):ϕ∗(3.2)
where the functor ϕ∗maps the subcategory of sheaves to the subcategory lim
←−I∈L Sh(Sk
I,D).
Finally, we introduce the sheafification functor
σ:DP(Sk
n)−→ Sh(Sk
n,D)
as the left adjoint of the inclusion. Here, we need to require the existence of pushouts in D;
this assumption is shown to be unnecessary below. Now (3.2) induces an adjunction,
σ◦ϕ!: lim
←−
I∈L
Sh(Sk
I,D)←→ Sh(Sk
n,D) :ϕ∗(3.3)
which we claim to be a pair of mutually inverse functors. In order to verify this, we show
that the unit and counit are isomorphisms. For the former, it suffices to show that, for every
sheaf G ∈ Sh(Sk
n,D) and every subset {∗, α} ⊆ Sk
nof cardinality 2, the unit morphism
(ϕ!ϕ∗G)({∗, α})−→ G({∗, α})
is invertible. We have
(ϕ!ϕ∗G)({∗, α})∼
=lim
−→
(I,U )∈LXop
P
α∈ρ×
I(U)
G(ρ×
I(U)).
According to Lemma 3.5 (1) below, the indexing category ϕop/{∗, α}of the colimit has a
final object (Iα,{∗, α|Iα}), with ρ×
Iα({∗, α|Iα}) = {∗, α}. This immediately implies the claim.
We proceed to prove that, for every object F ∈ lim
←−I∈L Sh(Sk
I,D), the counit morphism
F −→ ϕ∗σϕ!F
is invertible. Similarly as above, it suffices to show for all (J, {∗, β})∈LXPthat the map
F(J, {∗, β})−→ (ϕ∗σϕ!F)(J, {∗, β}) (3.4)
is an isomorphism in D. Using Lemma 3.5 (1), we compute the right-hand side as
(σϕ!F)(ρ×
J{∗, β})∼
=Y
α∈ρ−1
J(β)
(ϕ!F)({∗, α})∼
=Y
α∈ρ−1
J(β)
F(Iα,{∗, α|Iα}).
Then Lemma 3.5 (2) implies in particular that the map (3.4) is indeed an isomorphism. 
Lemma 3.5. In the terminology introduced in the proof of Theorem 3.4, let (J, V )be an
object of the category LXP. Then the following statements hold.
(1) Let α∈ρ×
J(V)r{∗}. There is a unique morphism
(Iα,{∗, α|Iα})−→ (J, V )
in LXP, where
Iα=[
αi<αi+1
{i, i + 1}.

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 13
(2) Let Fbe an object of lim
←−I∈L Sh(Sk
I,D)⊆ DLXP. There is an isomorphism
F(J, V )∼
−−→ Y
α∈ρ×
J(V)r{∗}
F(Iα,{∗, α|Iα}),
whose components are given by restriction along the unique morphisms from (1).
Proof. The even subsets of [n] of cardinality 2kare precisely the disjoint unions of ksubsets
of the form {i, i + 1}. Since α6≡ ∗, it follows that Iαis a (possibly non-disjoint) union of k
such subsets. However, it is contained in the even subset of [n] of cardinality 2kobtained
by inductively filling for each αi−1< αi< αi+1 either the maximal gap j < i of Iαor its
minimal gap j > i.
The key observation is that the subsets I∈ L which contain Iαare exactly those with
ρ×
I(ρI({∗, α})) = {∗, α}.
This implies that for a morphism in LXPof the form (Iα,{∗, α|Iα})←(I, U )→(J0, V 0),
we always have U={∗, α|I}. Thus, the only condition on V0is that α|J0∈V0, and we can
assume without loss of generality that V={∗, α|J}. In order to describe morphisms
µ: (Iα,{∗, α|Iα})−→ (J, {∗, α|J})
in LXP, we consider αas a sequence of kbars situated in a diagram of [n], signifying the
fact that αj< αj+1 by the bar j|(j+ 1). An object (I , {∗, α|I}) corresponds to marking the
elements i∈I⊆[n], and the zig-zag µis a sequence of moves which shift the markings. Each
move consists of adding and then removing certain markings (adhering to the constraints
imposed by the definition of LXP).
The object (Iα,{∗, α|Iα}) marks all elements adjacent to a bar; that is, we visualize it as
a diagram of the following exemplary form,
−−•|•−•|•|•−−•| . . . |•−−−
where ’•’ indicates a marked element and ’−’ an unmarked element of [n]. A single ’•’ at a
vertex i∈[n] between two bars (i.e., ’|•|’) corresponds precisely to the case αi−1< αi< αi+1
from above. Since α|J6≡ ∗, this implies that i∈J; in fact, this condition states exactly that
there is an element of Jin every region cut out by the bars.
In order to see that µis unique (if it exists), we first note that a ’•’ can never cross a bar.
Indeed, this would require some move
(I, {∗, α|I})←− (H, {∗, α|H})−→ X∈LXP
which adds a marking to some i∈Iα. Then we can define β∈ P(Sk
H) by β|Hr{i}≡α|Hr{i}
and by replacing the jump αi−1=αi< αi+1 with βi−1< βi=βi+1; but this contradicts the
requirement that the left leg of the move be cartesian.
Then uniqueness follows from the fact that moves which are constrained within different
sets of bars commute with one another, while the moves occuring between two particular
bars all compose to the same shift of markings.
For the existence of µ, we observe that after adding markings for each ’|•|’ as described
above (filling the gaps of Iα; where we can always choose the gap closest to an element of J),
we can remove at least one marking adjacent to each bar (with the exception of the ’| • |’, in
which case the vertex lies in Jalready, as we have seen).
Then we can move each ’•’ towards its intended position in Jby repeatedly marking the
adjacent vertex and removing the original; moreover, once a ’•’ has reached its destination,
we can duplicate it. This requires no further sets of the form {i, i + 1}to cover all markings,
that is, we stay within Lin this process (as of course J∈ L itself).
Finally, statement (2) follows from the above, since F(J, V ) = F(ρ×
JV), and similarly,
F(Iα,{∗, α|Iα}) = F({∗, α});
but condition (b) tells us that the restriction F|P(Sk
J)to the fibre π−1(J) is a sheaf. 

14 THOMAS POGUNTKE
Note that the n-cells of the 1-dimensional Segal construction constitute a pointed category
with finite products again. We write S(k)
⊕(D)∈Cat∆×kfor the k-fold iterate of Sh1i
⊕(D).
The following result not only provides another perspective on the higher Segal construction,
but together with Proposition 2.16, it yields an alternative proof of Theorem 3.4.
Theorem 3.6. Let k≥1. The k-dimensional Segal construction of a pointed category D
with finite products is naturally equivalent to the total simplicial object of its k-fold Segal
construction,
Shki
⊕(D)'
−−→ T S (k)
⊕(D).
Proof. For k= 1, there is nothing to show. By induction, it suffices to prove, for every k > 1,
Shki
⊕(D)'
−−→ T(Shk−1i
⊕(Sh1i
⊕(D))).
Therefore, we need to construct, for all k, n > 0, a natural equivalence of categories
Sh(Sk
n,D)'
−−→ lim
←−
I∪J=[n]
I≤J
Sh(Sk−1
I,Sh(S1
J,D)).(3.5)
Note that by Lemma 3.3, the right-hand side is computed by the 1-categorical limit. Now,
we first consider Sh(Sk−1
I,Sh(S1
J,D)) as a full subcategory of
Fun(P(Sk−1
I)op,Fun(P(S1
J)op,D))
'Fun(P(Sk−1
I)op × P(S1
J)op,D))
'Fun(P(Sk−1
IqS1
J)op,D)),
where qis the coproduct of pointed sets. Define the two maps λ, ρ :Sk
n→Sk−1
IqS1
Jby
λ(α) = α|Iif α(I)⊆[k−1],
∗otherwise, and ρ(α) = α|Jif α(J)⊆ {k−1, k},
∗otherwise.
Then we claim that an equivalence as in (3.5) descends from the induced functors
Sh(Sk
n,D)−→ Fun(P(Sk−1
IqS1
J)op,D)),F 7−→ (W7→ F(λ×W∩ρ×W)).
Firstly, let us show that the essential image of this functor lies in Sh(Sk−1
I,Sh(S1
J,D)). This
amounts to proving that for every U∈ P(Sk−1
I), the assignment V7→ F(λ×U∩ρ×V) is a
sheaf on P(S1
J). But this reduces straight-forwardly to the sheaf property of F, as
F(λ×U∩ρ×V)∼
−−→ Y
ε∈VY
α∈λ×U∩ρ×{∗,ε}
F({∗, α})∼
←−− Y
ε∈V
F(λ×U∩ρ×{∗, ε}).
The analogous argument shows that U7→ F(λ×U∩ρ×V) is a sheaf for every V∈ P(S1
J),
which yields the rest of the claim, since products of sheaves are computed point-wise.
Next, to see that these functors form into a map into the limit boils down to the transitivity
of restriction, (α|I)|IrJ=α|IrJfor I, J ⊆[n] as before.
Finally, to construct the inverse, given (FI,J )∈lim
←−I∪J=[n]
I≤J
Sh(Sk−1
I,Sh(S1
J,D)), we extend
{∗, α} 7−→ FI(α),J(α)({∗, α|I(α)})({∗, α|J(α)}),for I(α) = α−1[k−1], J (α) = {iα} q α−1(k),
where iα∈I(α) is the maximal element, to a sheaf on P(Sk
n). In one direction, we use
λ×{∗, α|I(α)} ∩ ρ×{∗, α|J(α)}={∗, α}
to see that the two constructions are inverse to one another. Conversely, let F ∈ Sh(Sk
n,D)
denote the image of some (FI,J ). For every (U, V )∈ P(Sk−1
I)× P(S1
J), I∩J6=∅, we get
F(λ×U∩ρ×V)∼
−−→ FI,J (U)(V)
via the following isomorphism,
Y
α∈λ×U∩ρ×V
FI(α),J(α)({∗, α|I(α)})({∗, α|J(α)})∼
−−→ Y
(β,ε)∈U×V
FI,J ({∗, β })({∗, ε}).

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 15
This map arises via the mutually inverse identifications α7→ (λ(α), ρ(α)) and (β, ε)7→ β∪ε
of the respective indexing sets, with its components given by the fact that
(FI,J )∈lim
←−
I∪J=[n]
I≤J
Sh(Sk−1
I,Sh(S1
J,D)),
which provides, for each α=β∪εas above, a chain of isomorphisms
FI(α),J(α)({∗, α|I(α)})({∗, α|J(α)})∼
=FI0,J0({∗, α|I0})({∗, α|J0})∼
=. . . ∼
=FI,J ({∗, β })({∗, ε}),
where I0=I(α)r{iα}, and J0={iα−1} q J(α). 
Definition 3.7. We denote by E'the maximal subgroupoid of a category E.
Corollary 3.8 (Delooping).Let k≥1, and let K⊕(D) = Ω|Sh1i
⊕(D)'|be the direct sum
K-theory space of D. There is a natural homotopy equivalence
Ωk|Shki
⊕(D)'|'
−−→ K⊕(D).(3.6)
Proof. By Theorem 3.6 as well as Lemma 3.9 below, this reduces to the case k= 1, which is
a special case of [25], Proposition 1.5. 
Lemma 3.9. Let Y∈Top∆×kbe a k-fold simplicial space. Then the natural map DY →T Y
induces an equivalence
|Y|'|DY |'
−−→ |T Y |.
Proof. Let us consider Y•,•∈(Set∆)∆×k. When k= 2, we obtain the claim from
|DY | ' |D([m]7→ DYm,•)|'
←−− |T([m]7→ DYm,•)|'
−−→ |T([m]7→ T Ym,•)|'|T Y |,
which holds by [5], Theorem 1.1 as well as §7; also see [26], Theorem 1. By iterating this
argument, the statement follows for all k.
4. The higher Waldhausen construction
We begin by recalling the following non-additive generalization of a Quillen exact category
from [11], Definition 2.4.2. For non-additive examples, see loc.cit., Example 2.4.4 ff, as well
as Example 5.3.
Definition 4.1. A proto-exact category Econsists of a pointed category Etogether with two
wide subcategories E,E⊇ E', of so-called admissible monomorphisms, resp. admissible
epimorphisms, denoted by BA, resp. DC, subject to the following conditions. All
the maps 0 Aand A0 are admissible. Moreover, a square of the form
B A
C D
(4.1)
is cocartesian if and only if it is cartesian, and a diagram CBA, resp. CDA,
can be completed to a square as in (4.1) by a pushout, resp. pullback.
A short exact sequence is a bicartesian square (4.1) with C= 0. A functor ω:E → B
between proto-exact categories is called exact if it preserves short exact sequences.
Now fix a proto-exact category E.
Definition 4.2. A morphism A→Bin Eis called admissible if it factors as the composition
A B
C
of an admissible epimorphism followed by an admissible monomorphism in E.

16 THOMAS POGUNTKE
Consider a sequence of admissible morphisms together with their corresponding (unique
up to unique isomorphism, by [4], Lemma 8.4) factorizations as above,
AkAk−1. . . A0.
CkCk−1. . . C1
The sequence Ak→Ak−1→. . . →A0will be called
•acyclic, if Ci+1 AiCiis a short exact sequence in Efor all 0 < i < k.
An acyclic sequence in Eas above is called
•left exact, if Ak→Ak−1is an admissible monomorphism (equivalently, Ak∼
−−→ Ck),
•right exact, if A1→A0is an admissible epimorphism (i.e., C1∼
−−→ A0),
•exact, if it is both left exact and right exact.
Definition 4.3. A proto-abelian category is a pointed category with kernels and cokernels,
the collections of which form a proto-exact structure.
Remark 4.4. In [9], Definition 1.2, proto-abelian categories are defined as pointed categories
on which the classes of all monomorphisms and epimorphisms define a proto-exact structure.
We have changed this terminology slightly, which will prove convenient later, rather than
introducing yet another different term. This does not exclude any of the main examples of
interest (like Example 5.3 (1)).
Let k, n ≥0. We write Fun([k],[n]) for the category of functors between the standard
ordinals [k] and [n], considered as small categories. Note that the objects of this category
correspond bijectively to the set of k-simplices of the simplicial set ∆n.
Definition 4.5. Let k≥0. For every n≥0, we define the category
S[k]
n(E)⊆Fun(Fun([k],[n]),E)
to be the full subcategory spanned by all diagrams Asatisfying the following conditions.
(a) For every (k−1)-simplex αin ∆n, we have
As∗
k−1α=. . . =As∗
0α= 0.
(b) For every (k+ 1)-simplex γin ∆n, the corresponding sequence
Ad∗
k+1γ−→ Ad∗
kγ−→ . . . −→ Ad∗
1γ−→ Ad∗
0γ
is acyclic.
We define Shk]
n(E), resp. S[ki
n(E), as the full subcategory of S[k]
n(E) on all Asuch that
(b0) For every (k+ 1)-simplex γin ∆n, the following sequence is left, resp. right, exact.
Ad∗
k+1γ−→ Ad∗
kγ−→ . . . −→ Ad∗
1γ−→ Ad∗
0γ
Finally, we introduce Shki
n(E)⊆S[k]
n(E) as the full subcategory of diagrams Awhich satisfy
(b00) For every (k+ 1)-simplex γin ∆n, the sequence
Ad∗
k+1γ−→ Ad∗
kγ−→ . . . −→ Ad∗
1γ−→ Ad∗
0γ
is exact.
By functoriality in [n], we obtain simplicial categories
S[k](E), Shk](E), S[ki(E), Shki(E)∈Cat∆.
We call Shki(E) the k-dimensional Waldhausen construction of E.

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 17
Remark 4.6. The (k+ 1)-skeleton of Shki(E) has an immediate description. Namely,
Shki
0(E)'. . . 'Shki
k−1(E)'0,and Shki
k(E)' E,
while Shki
k+1(E) is equivalent to the category of k-extensions in E. The dimensionality of the
Waldhausen construction also refers to the fact that the k-skeleton of |Shki(E)'|is equivalent
to the k-fold suspension Sk∧ |E'|, rather than the dimension of the diagrams it classifies.
Example 4.7. Let k≥0, and let Ebe a proto-exact category.
(1) For k= 0, the degeneracy condition (a) is empty, and therefore,
Sh0i(E)'NE(E')' E
is the nerve of the maximal subgroupoid of E, categorified by arbitrary morphisms
in E, which is equivalent to the constant object Eitself. Similarly, Sh0](E)'NE(E),
and dually, S[0i(E)'NE(E). Rather more subtly, S[0](E)'
−−→ NE(E) if and only if
Eis proto-abelian (in the sense of Definition 4.3), by [13], Proposition 3.1.
(2) For k= 1, we recover a version of the original construction Sh1i(E)'S(E) from [29],
whose n-cells are given by the category formed by strictly upper triangular diagrams
with bicartesian squares, as follows.
0A01 A02 · · · A0n
0A12 ...A1n
0....
.
.
0A(n−1)n
0



(4.2)
This is a refinement of Quillen’s foundational construction Q(E) in [22], which is the
correspondence category on Ewith morphisms of the form CBA. Namely, the
forgetful functor from the edgewise subdivision ES(E)→NE(Q(E)) is an equivalence,
that is, the whole diagram A∈S2n+1(E) of shape (4.2) is uniquely recovered from
A(n−1)(n+1) . . . A0(2n)
An(n+1) A(n−1)(n+2) . . . A1(2n)A0(2n+1)
. . . (4.3)
by taking successive pullbacks and pushouts in E.

18 THOMAS POGUNTKE
(3) For k= 2, the simplicial category Sh2i(E) was introduced by Hesselholt-Madsen [18].
An element A∈Sh2i
4(E) of its 4-cells is a diagram of the following form.
A012 A013 A014
A023 A024
A123 A124
A034
A134
A234
·y
p
·
(4.4)
Note that the middle square is neither cartesian nor cocartesian. Rather, the diagram
consists of bicartesian cubes (cf. Remark 4.8), as indicated in the following picture.
A012 A013 A014
0 0 0
0A023 A024
0A123 A124
0A034 0
0A134
0A234
(4.5)
Remark 4.8. In general, Shki(E) is composed of (k+ 1)-dimensional bicartesian hypercubes.
More precisely, let A∈S[k]
n(E). Then Alies in Shk]
n(E) if and only if
Aβ= lim
←−
β<β0
β0−β≤1
Aβ0(4.6)
for every k-simplex βin ∆nwith βk−i< n −ifor 0 ≤i≤k. Then Lemma 4.9 below implies
that the dual condition defines S[ki
n(E) inside S[k]
n(E), so that A∈S[ki
n(E) if and only if
lim
−→
β0<β
β−β0≤1
Aβ0=Aβ
for every k-simplex βin ∆nwith i<βifor all 0 ≤i≤k. Together, these yield the claim for
Shki
n(E)'Shk]
n(E)×S[k]
n(E)S[ki
n(E).
In order to see the first statement, note that the sequence of admissible morphisms Ad∗
•γ
corresponding to some (k+ 1)-simplex γin ∆ndefines a hypercube conv(Ad∗
•γ), formed by
all Aβwith d∗
k+1γ≤β≤d∗
0γ. If the maps (4.6) are isomorphisms, the minimal subhypercubes
of conv(Ad∗
•γ) are cartesian, and hence so is conv(Ad∗
•γ) as their composition. Therefore,
Ad∗
k+1γ= lim
←−
d∗
k+1γ <β≤d∗
0γ
Aβ= ker(Ad∗
kγ→Ad∗
k−1γ).

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 19
Conversely, let βbe a k-simplex in ∆nwith βk−i< n −ifor all 0 ≤i≤k. Then we can infer
inductively that the hypercube Qβon all β0≥βwith β0−β≤1 is cartesian. First, assume
that |β|=Pβiis maximal. Thus, β= (d∗
0)n−k∆n
n−1 = d∗
k+1(d∗
0)n−k−1∆n
n. But then Qβis
exactly given by the hypercube conv(Ad∗
•(d∗
0)n−k−1∆n
n).
In general, consider γ=βq {n}. Then d∗
k+1γ=β≤β+ 1 ≤d∗
0γ, and hence conv(Ad∗
•γ)
contains Qβentirely. But its complement is covered by hypercubes Qe
βwith |e
β|>|β|, which
are cartesian by induction. Therefore, so are all of their compositions, and thus so is Qβ.
The following observation makes the inherent symmetry by duality precise.
Lemma 4.9. The duality on ∆induces equivalences of simplicial categories
Shki(E)'
−−→ Shki(Eop ),
Shk](E)'
−−→ S[ki(Eop ),
S[k](E)'
−−→ S[k](Eop ).
Proof. This is immediate from the definitions. 
Lemma 4.10. Let k≥0,n≥1,0≤i≤n, and let Ebe a proto-exact category. The face
map
∂i:S[k]
n(E)→S[k]
n−1(E)
is an isofibration. In particular, the analogous statements hold for Shk](E)and S[ki(E), as
well as the higher Waldhausen construction of Eitself.
Proof. Let Φ : ∂i(A)∼
−−→ B0be an isomorphism in S[k]
n−1(E). We construct a lift B∈S[k]
n(E)
of B0as follows,
B: ([k]β
−→ [n]) 7−→ Aβif β /∈im(di)∗,
B0
αif β= (di)∗α.
The map in Bfor β≤e
βis given by the corresponding arrow in A, resp. B0, if β , e
β /∈im(di)∗,
resp. both β= (di)∗αand e
β= (di)∗eα. Otherwise, we define
(Bβ→Be
β) = ((Aβ→Ae
β
Φ
−→ B0
eα) if β /∈im(di)∗and e
β= (di)∗eα,
(B0
α
Φ
←− Aβ→Ae
β) if β= (di)∗αand e
β /∈im(di)∗.
The lifting of Φ itself is then straightforward. 
Remark 4.11. In particular, all the limits with transition maps given by compositions of
∂iwhich we consider throughout are computed by the respective 1-categorical limits; we will
make no further mention of this henceforth.
This also applies to Theorem 4.15, whose proof in turn can be used to provide an alternative
(needlessly complicated) argument for Lemma 4.10 by reducing to the classical case k= 1,
which further reduces to the case k= 0 via the equivalences Sh0]
n−1(E)'Sh1i
n(E)'S[0i
n−1(E)
from Lemma 4.12 below.
Lemma 4.12. The path spaces of the one-dimensional Waldhausen construction are given
by
PSh1i(E)'Sh0](E)'NE(E),and dually, PSh1i(E)'S[0i(E)'NE(E).
Proof (cf. [11], Lemma 2.4.9). Let n≥1. We construct an inverse to the forgetful functor
PSh1i
n−1(E)−→ S[0i
n−1(E).
Given A∈S[0i
n−1(E), we define b
A∈Sh1i
n(E)'PSh1i
n−1(E) as its right Kan extension along
[n−1] dn
−→ [n]−∪{n}
−−−−→ C −→ Fun([1],[n]),
where C={β∈Fun([1],[n]) |β0=β1or β1=n}is considered as a full subcategory.
Explicitly, this results in extending A∼
=(A0n. . . A(n−1)n) by zeroes on the diagonal
as in (4.2), then taking successive pullbacks to recover the whole Waldhausen cell. 

20 THOMAS POGUNTKE
We are now prepared to state our main result.
Theorem 4.13. Let Ebe a proto-exact category, and k≥0. The k-dimensional Waldhausen
construction Shki(E)is a 2k-Segal category.
Proof. The case k= 0 is settled by Example 4.7 (1). For k= 1, this is one of the main results
of [11], namely Proposition 2.4.8. As observed in op.cit., Example 6.3.3, we can apply the
path space criterion to Lemma 4.12 to conclude.
In general, this strategy only works under an additional assumption, as explained in §5.
Instead, the result follows from Theorem 4.15 below, together with Proposition 2.19. 
In light of Remark 2.21, another proof for k= 1 is provided by Example 4.7 (2), where we
have seen that ESh1i(E) is Segal.
An analogue of Theorem 4.13 in the context of stable ∞-categories is a result of work in
progress by Dyckerhoff and Jasso.
Definition 4.14. We write S(k)(E)∈Cat∆×kfor the k-fold iterate of the 1-dimensional
Waldhausen construction, where each Sh1i
n(E) carries the point-wise proto-exact structure.
Theorem 4.15. Let k≥0, and let Ebe a proto-exact category. There is a natural equivalence
of simplicial categories between the k-dimensional Waldhausen construction of Eand the total
simplicial object of its k-fold Waldhausen construction,
Shki(E)'
−−→ T S (k)(E).
Proof. For k≤1, this is tautological. By the same reasoning as in the proof of Theorem 3.6,
it is sufficient to construct, for all n, k ≥1, a natural equivalence of categories
Shki
n(E)'
−−→ lim
←−
I∪J=[n]
I≤J
Shk−1i
I(Sh1i
J(E)).
To define the functor, we use that the right-hand side is a full subcategory of
lim
←−
I∪J=[n]
I≤J
Fun(Fun([k−1], I),Fun(Fun([1], J ),E))
'lim
←−
I∪J=[n]
I≤J
Fun(Fun([k−1], I)×Fun([1], J),E)
'Fun( lim
−→
I∪J=[n]
I≤J
Fun([k−1], I )×Fun([1], J ),E).
Consider the following full subcategory of the indexing category for its elements,
{(α, ε)∈lim
−→
I∪J=[n]
I≤J
Fun([k−1], I )×Fun([1], J )|αk−1=ε0}.
This is equivalent to Fun([k],[n]) via (α, ε)7→ α∪ε, with inverse induced by the functor
Fun([k],[n]) −→ lim
−→
I∪J=[n]
I≤J
Fun([k−1], I )×Fun([1], J ), β 7−→ (β|[k−1], β|{k−1,k}).
Now let A∈Shki
n(E). Then its left Kan extension A!along the inclusion
Fun([k],[n]) −→ {(α, ε)∈lim
−→
I∪J=[n]
I≤J
Fun([k−1], I )×Fun([1], J )|αk−1=ε0or ε0=ε1}
amounts to an iterated extension by zeroes, as in the proof of Lemma 4.12. Then we define
the image b
A∈lim
←−
I∪J=[n]
I≤J
Shk−1i
I(Sh1i
J(E)) of Aas a further left Kan extension to the whole

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 21
indexing category lim
−→
I∪J=[n]
I≤J
Fun([k−1], I )×Fun([1], J ), which again is an iterated version of the
corresponding construction in the proof of Lemma 4.12. Note that the Kan extension exists,
since it only involves taking cokernels of admissible monomorphisms.
The construction is fully faithful by universality, and essential surjectivity is seen as follows.
For A0∈lim
←−I∪J=[n]
I≤J
Shk−1i
I(Sh1i
J(E)), a preimage A∈Shki
n(E) is given by glueing together the
sequence Ad∗
k+1γ→. . . →Ad∗
0γas A0
(γ0···γk−1,γk−1γkγk+1)×A0
(γ0···γk−1,γkγk+1)A0
(γ0···γk,γkγk+1),
which is meant to be short-hand notation for the following sequence,
A0
(γ0···γk−1,γk−1γk)→A0
(γ0···γk−1,γk−1γk+1)→A0
(γ0···γk−2γk,γkγk+1)→. . . →A0
(γ1···γk,γkγk+1).

Example 4.16. As a special case of Theorem 4.15, we illustrate the equivalence of categories
Sh2i
4(E)'S(2)
{0,1},{1,2,3,4}(E)×S(2)
{0,1},{2,3,4}(E)S(2)
{0,1,2},{2,3,4}(E)×S(2)
{0,1,2},{3,4}(E)S(2)
{0,1,2,3},{3,4}(E)
in the following diagram, where the glueing is represented by the two dashed sequences.
A123
A023 A124
•A024 •
• •
A013 A014 •A034 A134
A012 A234
Let k≥1. As can be seen either directly or with the help of Theorem 4.15, the functor
Φ: ExCat −→ Top∗,E 7−→ |Shk−1i(E)'|,(4.7)
satisfies the hypotheses of [17], §1.3. This permits us to draw the following consequences on
geometric realizations.
Corollary 4.17 (Additivity).Let Ebe an exact category, and k≥1. Then the map
|Shki(S2(E))'|'
−−→ |Shki(E)'|×|Shki(E)'|
induced by the functor (∂2, ∂0) : S2(E)→ E × E is a weak equivalence. In fact, the simplicial
space |Shki(S•(E))'|is a Segal space.
Proof. By Theorem 4.15, this is precisely [17], Theorem 1.3.5 (2), with Φ as in (4.7). 
Needless to say, the other versions of additivity in [17], Theorem 1.3.5, hold as well; this
also allows us to deduce the following as in loc.cit. via the proof of [20], Proposition 3.6.2.
Corollary 4.18 (Delooping).Let k≥1, and let K(E)denote the algebraic K-theory space
of the exact category E. There is a natural homotopy equivalence
Ωk|Shki(E)'|'
−−→ K(E).

22 THOMAS POGUNTKE
Remark 4.19. Similarly to Corollary 3.8, we can also use Theorem 4.15 to immediately
reduce to the case k= 1 ([29], Theorem 1.4.2 and Proposition 1.3.2, resp. Proposition 1.5.3).
In conclusion, the algebraic K-theory spectrum of Eis given by the sequence of maps
|Sh0i(E)'| −→ Ω|Sh1i(E)'| −→ Ω2|Sh2i(E)'| −→ . . . ,
where, for each k≥0, the morphism
|Shki(E)'| −→ Ω|Shk+1i(E)'|
is induced by the inclusions of the cells θn:Shki
n(E)−→ Shk+1i
n+1 (E) as Sh1i
1(Shki
n(E)), extended
by zeroes appropriately, for all n≥0. Namely, [29], Lemma 1.5.2, constructs the map via
S(k)(E)−→ PS(k+1)(E)−→ S(k+1)(E),(4.8)
inducing a map S(k)(E)→LS(k+1)(E) to the simplicial loop space. Totalizing (4.8) yields
Shki(E)θ
−−→ PShk+1i(E)−→ Shk+1i(E)
by Lemma 2.18. As before, this produces the desired map Shki(E)−→ LShk+1i(E).
Finally, the following result can be understood as a refinement of the delooping theorem
in that morally it states that Shki(E) exhibits the Ek-algebra structure on K(E).
Corollary 4.20. Let Ebe an exact category, and k≥1. The simplicial space K(Shki(E)) is
a lower (2k−1)-Segal space.
Proof. From Theorem 4.15 as well as Lemma 4.21 below, we deduce that
K(Shki(E)) 'K(T S(k)(E)) 'T K (S(k)(E)).
But by the additivity theorem, K(S(k)(E)) is a Segal space in every direction, and therefore,
the statement follows from Proposition 2.16. 
Lemma 4.21. Let E,B,Dbe proto-exact categories, let ω:E → B be an exact isofibration,
and let ν:D → B be an exact functor. Then
S(E ×BD)'
−−→ S(E)×S(B)S(D),
and the right-hand side agrees with the 1-categorical fibre product. In particular,
K(E ×BD)'
−−→ K(E)×K(B)K(D).
Proof. Firstly, the functor Sn(ω) is an isofibration, for all n∈N. Indeed, this can be proven
inductively (cf. [29], Lemma 1.6.6). The case n= 1 is our assumption. Given an isomorphism
ω(M0)· · · ω(Mn−1)ω(Mn)
V0· · · Vn−1Vn,
∼ ∼ f
∼
in Sn+1(B)'Sh0]
n(B), for n≥1, we can lift all but the last vertical arrow f, and subsequently
take the pushout
M0· · · Mn−1Mn
N0· · · Nn−1N.
∼∼p
·ϕ
Then ϕis a lift of f. This shows the second part of the first claim. Note that we use
repeatedly that all functors involved are exact. Now, applying Sn+1(−) to the projections
yields a unique functor Sn+1(E ×BD)→Sn+1 (E)×Sn+1(B)Sn+1 (D). But under Lemma 4.12
again, this becomes the equivalence
Sh0]
n(E ×BD) = NE×BD
n((E ×BD))'NE
n(E)×NB
n(B)ND
n(D) = Sh0]
n(E)×Sh0]
n(B)Sh0]
n(D)

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 23
since by definition, (E ×BD)=E×BD, and because
Nn(E ×BD) = Fun([n],E ×BD)'Fun([n],E)×Fun([n],B)Fun([n],D) = Nn(E)×Nn(B)Nn(D).
Finally, for the last claim, (−)'and Ω are both right adjoints, while geometric realization
commutes with fibre products by [7]. 
5. Stringent categories and path spaces
Let Ebe a proto-exact category. In this section, we investigate which further higher Segal
conditions the higher dimensional Waldhausen construction of Eand its variants satisfy under
the following additional homological algebraic assumption on E.
Definition 5.1. A stringent category is a proto-exact category Esuch that for every triangle
A B
C
f
g◦f
g
of admissible morphisms in E, the induced sequence of maps
ker(f)−→ ker(g◦f)−→ ker(g)−→ coker(f)−→ coker(g◦f)−→ coker(g)
is exact.
Remark 5.2. A proto-exact category Eis stringent if and only if the functor
PPSh2i(E)−→ S[0](E), A 7−→ A[0]⊕−⊕[0],
defines an equivalence of simplicial categories. Indeed, the equivalence
PPSh2i
2(E)'
−−→ S[0]
2(E)
is precisely the definition of stringency, while the converse is part of Proposition 5.11.
Example 5.3. By symmetry of the definition, Eis stringent if and only if Eop is. If Eis
stringent, then so is Fun(Q, E), for a small category Q.
(1) In particular, the category Fun(Q, vectF1) of representations of any small category Q
in (finite) F1-vector spaces is stringent, where vectF1is the wide subcategory of Fin×
on all f:V→Wwhich are injective outside of the kernel f−1(∗).
(2) Consider the pointed category Eon a non-zero object Vwith End(V) = {0,1, ε},
such that ε2= 0. It can easily be verified directly that εadmits neither a kernel nor
a cokernel, and thus Eis stringent.
If Fis a field, then the F-linear Cauchy completion of Eis an additive stringent
category, namely the category of finite free F[x]/(x2)-modules.
Remark 5.4. An additive category is stringent if and only if it is weakly idempotent complete
in the sense of [4], Definition 7.2, or equivalently, any of the characterizations given there,
like the so-called cancellation property loc.cit., Corollary 7.7.
This is one of Heller’s axioms (cf. op.cit., Proposition B.1), the rest of which also follow
from weak idempotent completeness. In our setting, the entirety of these axioms (with the
obvious exception of additivity) still provides an equivalent characterization of stringency.
Indeed, the proof of op.cit., Proposition 8.11, does not require additivity, and the converse is
shown in loc.cit., Lemma 8.12, as well as Proposition 5.5 below.
As a consequence, the snake lemma holds in any stringent category E. The neat argument
presented in loc.cit., Corollary 8.13, does not quite apply here (as it ultimately relies on the
additive structure of an exact category); however, the proof of [16], Proposition 4.3, does.

24 THOMAS POGUNTKE
Proposition 5.5. Let Ebe a stringent category, and consider a diagram of the form
A0B0C0
A B C
A00 B00 C00
in E, where all rows as well as all columns but the first are exact. Then A0AA00 is a
short exact sequence as well.
Proof. The kernel-cokernel sequence for the lower triangle (of admissible maps) in the square
B0C0
B C
is given by 0 −→ A0−→ A−→ B00 −→ C00 −→ 0, and ker(B00 →C00) = A00.

We are now prepared to state the first main result of this section.
Proposition 5.6. Suppose Eis a stringent category. Then the simplicial categories Shk](E)
and S[ki(E)are lower (2k−1)-Segal.
Remark 5.7. When k≥2, the assumption in Proposition 5.6 that Ebe stringent is necessary,
which is illustrated by the following observation, at least in the additive case. Suppose Eis
not weakly idempotent complete. Then Sh1](E) is not lower 3-Segal.
Indeed, there exist Af
−→ Bg
−→ Cin Esuch that gand g◦fare admissible monomorphisms
but fis not. However, by [4], Remark 8.2, it can be written as a composition of strict maps
A(1,f)
−−−→ A⊕B(0,1)
−−−→ B.
Now suppose fadmits a kernel Dand consider the following possible element of Sh1]
4(E).
0 0 D D A
0A A A ⊕B
0 0 B
0C
0
·y·y·y
(1,f)
·y·y
(0,1)
·y
g
(5.1)
Then the triple of diagrams
0D D 0D A A A A ⊕B
0A A 0A A ⊕B0 0 B
0 0 0 C0C
·y·y·y·y(1,f)·y·y(0,1)
·y·y(0,g)·yg

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 25
defines an element in the right-hand side of the lower 3-Segal map for Sh1]
4(E). However, it
does not lie in its essential image, because the sequence
D A B
f
indexed by {0,2,4}is not left exact (the map fnot being strict), so (5.1) /∈Sh1]
4(E).
Similarly, the dual argument shows that the 4-cells of S[1i(E) do not satisfy the lower
3-Segal condition (by Lemma 4.9).
Example 5.8. Let us illustrate the lowest 3-Segal conditions for Sh2i(E), which is more
conveniently done by depicting an element of its 4-cells as the following projection of (4.4).
A123
A023 A124
A024
A013 A014 A034 A134
A012 A234
(5.2)
The dashed part marks the image of (5.2) in the right-hand side of the upper 3-Segal map
Sh2i
4(E)−→ Sh2i
3(E)×Sh2i
2(E)Sh2i
3(E).
The upper 3-Segal condition says that the whole diagram (5.2) is uniquely recovered from the
dashed subdiagram. Note that the complementary statement (for the lower 3-Segal map) is
false in general. In fact, it is equivalent to uniquely filling the frame of short exact sequences
C4A023 A123
C3A0
024 A124
C2C1C0
(5.3)
where Ci= coker(Ad∗
3d∗
i∆4
4
Ad∗
2d∗
i∆4
4)∼
=ker(Ad∗
1d∗
i∆4
4
Ad∗
0d∗
i∆4
4). However, there is an
obstruction to this, which is parametrized by the quotient groupoid
[Ext1(C0, C4)/Hom(C0, C4)],
as calculated in [9], Lemma 2.30 and Proposition 2.38, assuming that Eis abelian.
Our next observation will prove essential for our inductive arguments.
Proposition 5.9 (Hyperplane lemma).Let 1≤k≤l < m ≤n, and let Ebe a stringent
category. Then there is a natural functor
η
lm :Shk]
n(E)−→ Shk−1]
l(E), A 7−→ (β7→ coker(Aβ∪{l}Aβ∪{m})).
Dually, there is a corresponding natural functor
η
lm :S[ki
n(E)−→ S[k−1i
l(E), A 7−→ (β7→ ker(A{n−m}∪βA{n−l}∪β)).
Moreover, both of these restrict to functors on the higher Waldhausen construction,
Shki
n(E)Shk−1i
l(E).
η
lm
η
lm

26 THOMAS POGUNTKE
Proof. Let γbe a k-simplex in ∆l, and γ0=γq {m}. If l∈γ, then the sequence η
lm(A)d∗
•γ
is given by
coker(Ad∗
k+1γ0Ad∗
kγ0)Ad∗
k−1γ0Ad∗
k−2γ0. . . Ad∗
0γ0
which of course is indeed left exact. Furthermore, it is exact if and only if Alies in Shki
n(E),
by definition.
Now assume that l /∈γ. Then, for each vertex 0 < i < k, let us write
(Ad∗
i+1γ∪{l})Bi+1 Ad∗
iγ∪{l}Bi(Ad∗
i−1γ∪{l})
(Ad∗
i+1γ∪{m})Ci+1 Ad∗
iγ∪{m}Ci(Ad∗
i−1γ∪{m})
for the corresponding short exact sequence. Taking cokernels yields a diagram as follows.
Bi+1 Ci+1 Di+1
Ad∗
iγ∪{l}Ad∗
iγ∪{m}η
lm(A)d∗
iγ
BiCiDi
By the snake lemma, the right vertical sequence is short exact. Note that if A∈Shki
n(E),
B1∼
−−→ Ad∗
0γ∪{l}and C1∼
−−→ Ad∗
0γ∪{m}
which immediately implies also D1∼
−−→ η
lm(A)d∗
0γby definition. It remains to prove that
η
lm(A)d∗
kγ−→ η
lm(A)d∗
k−1γ
is an admissible monomorphism. In order to see this, we may show that the diagram
Ad∗
kγ∪{l}Ad∗
kγ∪{m}
Ad∗
k−1γ∪{l}Ad∗
k−1γ∪{m}
0Ad∗
k−1d∗
kγ∪{m−1,m}
(5.4)
is cartesian, by Lemma 5.10 below. In fact, we claim that it is the composition of pullbacks
Ad∗
kγ∪{l}Ad∗
kγ∪{l+1}. . . Ad∗
kγ∪{m}
Ad∗
k−1γ∪{l}Ad∗
k−1γ∪{l+1}. . . Ad∗
k−1γ∪{m}
0Ad∗
k−1d∗
kγ∪{l,l+1}. . . Ad∗
k−1d∗
kγ∪{m−1,m}.

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 27
To prove this claim, for each l≤j < m, we have the diagram
Ad∗
kγ∪{j}Ad∗
kγ∪{j+1}
Ad∗
k−1γ∪{j}Ad∗
k−1γ∪{j+1}
0Ad∗
k−1d∗
kγ∪{j,j+1}
in which the lower and outer rectangles are pullback, and therefore, so is the upper.
Finally, the functor η
lm is given by the map η
lm for Eop, via Lemma 4.9. 
Lemma 5.10. Let Ebe a stringent category, and consider a pullback diagram of admissible
morphisms in Eof the following form.
B1A1
B0A0
0D
·y
·y
(5.5)
The induced map C1C0is an admissible monomorphism, where Ci= coker(BiAi).
Proof. Since A0→Dand A1→Dare admissible, with kernels B0and B1, respectively, their
unique epi-mono factorizations are through C0, resp. C1, and we can extend the diagram to
B1A1C1.
B0A0C0
0D
·y
·y
Then the statement results from cancellation. 
The following result constitutes a generalization of Lemma 4.12.
Proposition 5.11. Let k≥1, and assume that Eis a stringent category. Then there are
equivalences of simplicial categories
PShki(E)'
−−→ Shk−1] (E), A 7−→ A[0]⊕−,
PShki(E)'
−−→ S[k−1i(E), A 7−→ A−⊕[0],
induced by the forgetful functors. For k≥2, there is an equivalence
PPShki(E)'
−−→ S[k−2] (E), A 7−→ A[0]⊕−⊕[0].
Proof. We prove the second statement first. For a diagram A∈S[k−1i
n(E), we construct its
image in PShki
n(E)'Shki
n+1(E) under the inverse functor as a right Kan extension. Namely,

28 THOMAS POGUNTKE
we extend by zero appropriately, and then into the kth dimension, as follows.
Fun([k−1],[n]) E
Fun([k−1],[n+ 1])
Cyl(ι|∆n
k−1)
Fun([k],[n+ 1])
A
ι
A!
λ
b
A(5.6)
Here, we have set ι= (dn+1 )∗, and the category Cyl(ι|∆n
k−1) is the cograph of its restriction
to the skeleton. The functor λis defined by (sk−1)∗on ∆n
k−1, and on Fun([k−1],[n+ 1]), it
maps
α7→ α∪ {n+ 1}.
Explicitly, b
Ais given by the diagram
β7−→ lim
←−
β≤λ(α)
A!
α∼
=Aβr{n+1}if n+ 1 ∈β,
ker(Ad∗
kβ→Ad∗
k−1β) otherwise.
Indeed, d∗
kβis initial amongst those objects of the indexing category of the limit which come
from Fun([k−1],[n+ 1]). If n+ 1 ∈β, then this is the only contribution. Otherwise, there
are additionally the objects of the form
[α]∈∆n
k−1with d∗
k−1β≤α.
Therefore, in that case, the limit reduces to just the pullback
b
Aβ∼
=lim
←−







A!
d∗
kβ
A!
[d∗
k−1β]A!
d∗
k−1β







∼
=lim
←− 





Ad∗
kβ
0Ad∗
k−1β





= ker(Ad∗
kβ→Ad∗
k−1β).
Now let γbe a (k+ 1)-simplex in ∆n+1. We claim that the corresponding sequence b
Ad∗
•γis
exact. If n+ 1 ∈γ, then this is simply given by
ker(Ad∗
kd∗
k+1γ→Ad∗
k−1d∗
k+1γ)Ad∗
kγr{n+1}. . . Ad∗
1γr{n+1}Ad∗
0γr{n+1}
which is an exact sequence in Eby definition.

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 29
Otherwise, the relevant sequence is given by
ker(Ad∗
kd∗
k+1γ→Ad∗
k−1d∗
k+1γ) ker(Ad∗
kd∗
k+1γ→Ad∗
k−1d∗
k+1γ)
ker(Ad∗
kd∗
kγ→Ad∗
k−1d∗
kγ) ker(Ad∗
kd∗
k+1γ→Ad∗
k−1d∗
kγ)
ker(Ad∗
kd∗
k−1γ→Ad∗
k−1d∗
k−1γ) ker(Ad∗
k−1d∗
k+1γ→Ad∗
k−1d∗
kγ)
ker(Ad∗
kd∗
k−2γ→Ad∗
k−1d∗
k−2γ) ker(Ad∗
k−2d∗
k+1γ→Ad∗
k−2d∗
kγ)
.
.
..
.
.
ker(Ad∗
kd∗
0γ→Ad∗
k−1d∗
0γ) ker(Ad∗
0d∗
k+1γ→Ad∗
0d∗
kγ).
or equivalently,
We prove exactness inductively. The first part of the sequence fits into a diagram of the form
b
Ad∗
k+1γb
Ad∗
kγAd∗
kd∗
k+1γ
0b
Ad∗
k−1γAd∗
k−1d∗
k+1γ
0Ad∗
k−1d∗
kγ.
·y·y
·y
By definition, the bottom left square is pullback, so we can pull it back to the top and then
to the left, since each outer rectangle is a pullback square by construction. Thus,
b
Ad∗
k+1γb
Ad∗
kγb
Ad∗
k−1γ
is a left exact sequence. Now, for each 0 < i < k, let us write
(Ad∗
i+1d∗
k+1γ)Bi+1 Ad∗
id∗
k+1γBi(Ad∗
i−1d∗
k+1γ)
(Ad∗
i+1d∗
kγ)Ci+1 Ad∗
id∗
kγCi(Ad∗
i−1d∗
kγ)
(5.7)
for the corresponding short exact sequences at the ith vertex of d∗
k+1γand d∗
kγ, respectively.
First, we show that Bk→Ckis an admissible epimorphism. But we have
Bk= coker( b
Ad∗
k+1γAd∗
kd∗
k+1γ),and Ck= coker( b
Ad∗
kγAd∗
kd∗
kγ).
Therefore, they fit into a diagram of the following form, which yields the claim.
b
Ad∗
k+1γb
Ad∗
kγAd∗
kd∗
k+1γ
0B0
kBk
0Ck
p
·p
·
p
·

30 THOMAS POGUNTKE
In particular, B0
k= ker(BkCk). Next, we show that Bk−1→Ck−1admits a kernel B0
k−1
in E. In fact, consider the following diagram.
BkAd∗
k−1d∗
k+1γBk−1
CkE D
CkAd∗
k−1d∗
kγCk−1
0C0
k−1C0
k−1
(3)
(1)
=
(10)
(2)
(20)
=
We have (1) by Remark 5.4, and (10) is its cokernel. The snake lemma yields (2) and (20),
and (3) is obtained dually to (1).
Now the snake lemma implies that the top row of the following diagram is short exact.
B0
kb
Ad∗
k−1γB0
k−1
BkAd∗
k−1d∗
k+1γBk−1
CkAd∗
k−1d∗
kγCk−1
In particular, this settles the case k= 2. For k≥3, we can rewrite the diagram
B2Ad∗
1d∗
k+1γB1=Ad∗
0d∗
k+1γ
C2Ad∗
1d∗
kγC1=Ad∗
0d∗
kγ
in terms of the hyperplane lemma (Proposition 5.9), namely as the upper part of the short
exact sequence of acyclic sequences
η
(n−γ1)(n−γ0)(A)d∗
k−1αA{γ0}∪d∗
k−1αA{γ1}∪d∗
k−1α
η
(n−γ1)(n−γ0)(A)d∗
k−2αA{γ0}∪d∗
k−2αA{γ1}∪d∗
k−2α
η
(n−γ1)(n−γ0)(A)d∗
k−3αA{γ0}∪d∗
k−3αA{γ1}∪d∗
k−3α
.
.
..
.
..
.
.
(5.8)
where α=d∗
0d∗
1γ. In particular, B2→C2is an admissible morphism. Applying the snake
lemma to the third morphism of short exact sequences in (5.8) tells us that the map
C0
2= coker(B2→C2)−→ coker(Ad∗
1d∗
k+1γ→Ad∗
1d∗
kγ)
is an admissible monomorphism, and therefore, by applying it to the first, that b
Ad∗
1γb
Ad∗
0γ.

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 31
Finally, we can iterate the argument, realizing (5.7) as the upper part of the diagram
η(i)(A)d∗
k−iα(i)A{γ0,...,γi−1}∪d∗
k−iα(i)η(i−1)(A)d∗
k−i+1α(i−1)
η(i)(A)d∗
k−i−1α(i)A{γ0,...,γi−1}∪d∗
k−i−1α(i)η(i−1)(A)d∗
k−iα(i−1)
η(i)(A)d∗
k−i−2α(i)A{γ0,...,γi−1}∪d∗
k−i−2α(i)η(i−1)(A)d∗
k−i−1α(i−1)
.
.
..
.
..
.
.
where η(i)=η
(n−γi)(n−γi−1)◦. . . ◦η
(n−γ1)(n−γ0)and α(i)=d∗
0. . . d∗
iγ. Then the sequence
B0
i+1 b
Ad∗
iγB0
i
is the beginning of the corresponding long exact snake, where B0
i= ker(Bi→Ci), and is
therefore a short exact sequence, as above.
Finally, the equivalence PShki(E)'
−−→ Shk−1](E) follows via Lemma 4.9 from the one
we have proven above. Furthermore, if k≥2, we obtain PPShki(E)'
−−→ S[k−2](E) as an
immediate consequence of the two. Namely, let A∈S[k−2]
n(E). Then the left Kan extension
analogous to (5.6) produces a diagram
b
A∈S[k−1i
n+1 (E) = PS[k−1i
n(E)'PPShki
n(E),
as all arguments above apply verbatim to show that b
Aconsists of right exact sequences. 
Remark 5.12. Proposition 5.11 can be seen as a higher analogue of the third isomorphism
theorem, in that the equivalence of categories Shk−1]
k+1 (E)'
−−→ PShki
k+1(E) = Shki
k+2(E) boils
down to the following statement. Given a configuration of left exact sequences of the form
Ak+1
kAk+1
k−1Ak+1
1Ak+1
0
Ak
k−1
Ak
1
Ak
0
Ak−1
1
Ak−1
0
A0
0

32 THOMAS POGUNTKE
where Aj
i=Ad∗
id∗
j∆k+1
k+1 in the previous notation, the induced maps between the cokernels
coker(Ak+1
1→Ak+1
0)−→ coker(Ak
1→Ak
0)−→ . . . −→ coker(A0
1→A0
0)
constitute an exact sequence in E.
Remark 5.13. Let us use the observation in Remark 5.12 to illustrate that the stringency
assumption on Ein Proposition 5.11 is necessary. For this, consider the dual situation of
Remark 5.7, so that both fand the composition Af
−→ Bg
−→ Care admissible epimorphisms
but gis not. Now let B0A, resp. C0A, be the kernel of f, resp. g◦f. If h:B0→C0
is admissible,
0 0 B0
0 0 C0
0A
·y·y
h
·y
lies in Sh1]
3(E), but it cannot be extended to PSh2i
3(E) since gis not admissible. If hitself is
not admissible either but admits a kernel D, we can construct a similar element
D D D ⊕B0
0B0B0⊕C0
0A
·y·y
0 1
0h


h
·y
(0,1)
of Sh1]
3(E). The corresponding sequence of cokernels B0h
−→ C0→Bg
−→ Cis again not exact.
Corollary 5.14. If Eis stringent, there are natural equivalences of simplicial categories
Shk](E)'
−−→ T(Sh0] S(k)(E)),
S[ki(E)'
−−→ T(S(k)S[0i(E)),
S[k](E)'
−−→ T(Sh0] S(k)S[0i(E)).
Proof. This is an immediate consequence of Proposition 5.11 and Theorem 4.15, as
Shk](E)'PShk+1i(E)'PT S(k+1) (E)'T P S(k+1)(E)'T(Sh0] S(k)(E)),
by Lemma 2.18. 
Theorem 5.15. Let Ebe a proto-exact category. The two-dimensional Waldhausen con-
struction Sh2i(E)is an upper 3-Segal category if and only if Eis proto-abelian.
Proof. By the path space criterion, it suffices to show that PPSh2i(E) is Segal. But Propo-
sition 5.11 below shows that for all n≥2, the forgetful functor
PPSh2i
n−2(E)−→ S[0]
n−2(E), A 7−→ (A01n→A02n→ · · · → A0(n−1)n),
is an equivalence of categories, identifying the double path space PPSh2i(E)'
−−→ NE(E)
with the categorified nerve of E, cf. Example 4.7 (1).
Conversely, the Segal condition for PPSh2i(E)'S[0](E) requires admissible morphisms
in Ebe closed under composition, hence Emust be proto-abelian already. 

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 33
Remark 5.16. Suppose Eis additive. Then Theorem 5.15 does not generalize to the higher
dimensional Waldhausen constructions, that is, Shki(E) is not upper (2k−1)-Segal for k6= 2.
Indeed, the diagram
0 0 0 A A
0A A ⊕A A
0A0
0 0
0
=
(0,1) =
(1,0)
(1,1)
(0,1)
(5.9)
is an element of the right-hand side of the lower 3-Segal map for PPSh3i
4(E), but does not
lie in its essential image.
We are now prepared to prove our main result. In particular, by Proposition 5.11 as well
as Lemma 4.9, the path space criterion provides a new proof of Theorem 4.13, under the
additional assumption that Ebe stringent.
Proof of Proposition 5.6. Let n≥2k. We show inductively that the lower (2k−1)-Segal map
Shk−1]
n(E)−→ lim
←−
I∈L([n],2k−1)
Shk−1]
I(E) (5.10)
is an equivalence. Throughout, for 0 ≤i≤n, let δirefer to the ith face map of ∆n
n, even
when applied to any subsimplex of it. That is, δ∗
iremoves the vertex i, and b
δ∗
iadjoins it.
First, consider the case n= 2k. By Lemma 5.17, the only k-simplex in ∆2knot contained
in an even subset of [2k] of cardinality 2kalready is
ε={0,2,...,2k}=δ∗
2k−1δ∗
2k−3· · · δ∗
1∆2k
2k.
But if Alies in the right-hand side of (5.10), then we can form the unique compositions
Aδ∗
2kεAδ∗
2k−2ε. . . Aδ∗
0ε
Aδ∗
2kb
δ∗
2k−1δ∗
2k−2εAδ∗
2k−2b
δ∗
2k−3δ∗
2k−4ε. . . Aδ∗
2b
δ∗
1δ∗
0ε
completing Ato an element of Shk−1]
2k(E). It remains to be shown that the resulting sequence
Aδ∗
2kε−→ Aδ∗
2k−2ε−→ . . . −→ Aδ∗
0ε
is left exact. We proceed by induction. The case k= 2 is settled by Theorem 5.15. In general,
since Aδ∗
2kεAδ∗
2k−2εis an admissible monomorphism (as a composition of such), it suffices
to prove that
coker(Aδ∗
2kεAδ∗
2k−2ε)−→ Aδ∗
2k−4ε−→ . . . −→ Aδ∗
0ε(5.11)
is a left exact sequence. For this, we use the hyperplane lemma. Namely, the functor
η
(2k−2)2k:Shk−1]
2k(E)−→ Shk−2]
2k−2(E)

34 THOMAS POGUNTKE
constructed in Proposition 5.9 is compatible with the corresponding lower Segal maps on
both sides, in that it induces a commutative diagram of the following form.
Shk−1]
2k(E) lim
←−
I∈L([2k],2k−1)
Shk−1]
I(E)
Shk−2]
2k−2(E) lim
←−
J∈L([2k−2],2k−3)
Shk−2]
J(E)
η
(2k−2)2kη
(2k−2)2k
Indeed, this is because we have d∗
0d∗
0ε=δ∗
2k−3δ∗
2k−5· · · δ∗
1∆2k−2
2k−2. But then, by induction, the
lower horizontal map is an equivalence, which by the above means precisely that (5.11) is a
left exact sequence.
In order to prove the Segal conditions for the higher cells Shk−1]
n(E), we once again employ
induction, now on the dimension n. If Alies in the right-hand side of the lower (2k−1)-Segal
map for Shk−1]
n(E), we first need to see that taking compositions completes Ato a well-defined
diagram of shape Fun([k−1],[n]) in E.
By Lemma 5.17, we need only consider sequences indexed by simplices γwith all vertices
separated by gaps. For gaps i∈[n] of size 1, there is (as before) a unique composition,
Aδ∗
i+1γAδ∗
i−1γ.
Aδ∗
i+1b
δ∗
iδ∗
i−1γ
For a gap of γof size l+ 1, say {i, . . . , i +l} ⊆ [n], each 0 ≤j≤ldefines the composition
Aδ∗
i+l+1γAδ∗
i−1γ.
Aδ∗
i+l+1b
δ∗
i+jδ∗
i−1γ
By induction, all possible compositions can be reduced to one of these. On the other hand,
they all agree, since for all 0 ≤j < j0≤l, the following diagram commutes.
Aδ∗
i+l+1γAδ∗
i+l+1b
δ∗
i+j0δ∗
i−1γ
Aδ∗
i+l+1b
δ∗
i+jδ∗
i−1γAδ∗
i−1γ
Finally, we apply induction to obtain the remaining exactness conditions for the completed
diagram of A. Namely, the sequence indexed by γis left exact, since ∂i(A) lies in the right-
hand side of the lower (2k−1)-Segal map of Shk−1]
n−1(E), for any gap iof γ.
Lemma 5.17. Let n≥2k. Let γbe a k-subsimplex of ∆nwith a pair of adjacent simplices.
Then γis contained in an even subset I⊆[n]of cardinality #I= 2k.
Proof. For n= 2k, this is clear. For the induction step, we can assume that n∈γ, otherwise
the statement follows tautologically from the induction hypothesis. Let 0 < m < n be the
maximal gap of γ. By induction, (γ∪ {m})r{n}is contained in an even subset I0⊆[n−1]
with #I0= 2k. But then γis contained in I= (I0r{m})∪ {n}, which is even in [n]. 

HIGHER SEGAL STRUCTURES IN ALGEBRAIC K-THEORY 35
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Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn, Germany
E-mail address:thomas.poguntke@hcm.uni-bonn.de