Camell Kachour

Université Paris-Sud 11 | Paris 11 · Département de Mathématiques

In this article 1 we show how to build aspects of articles [24, 25, 34] but with the cubical geometry. Thus we define a monad on the category CSets of cubical sets which algebras are models of cubical weak ∞-categories. Also for each n ∈ N we define a monad on CSets which algebras are models of cubical weak (∞, n)-categories. And finally we define...

This article describe globular weak $(n,\infty)$-transformations ($n\in\mathbb{N}$) in the sense of Grothendieck, i.e for each $n\in\mathbb{N}$ we build a coherator $\Theta^{\infty}_{\mathbb{M}^n}$ which sets models are globular weak $(n,\infty)$-transformations. A natural globular filtration emerges from these coherators.

In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $\Theta^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $\Theta_0$ whose o...

In these notes we describe models of globular weak (∞, m)-categories (m ∈ N) in the Grothendieck style, i.e for each m ∈ N we define a globular coherator Θ ∞ M m whose set-models are globular weak (∞, m)-categories. Then we describe the combinatorics of the small category Θ0 whose objects are cubical pasting diagrams and whose morphisms are morphis...

In the second part of this article we use the cubical operad B 0 C of cubical weak ∞-categories (built in [10]) as a fundamental step to associate to any topological space X its fundamental cubical weak ∞-groupoids Π∞(X), and this endows a functor Top ∞-CGrp Π∞(−) which has a left adjoint functor CN∞. This pair of adjunction (CN∞, Π∞(−)) should put...

In this article, divided in two parts, we show how to build main aspects of the article [1] but with the cubical geometry. This first part is devoted to build the contractible S-operad B 0 C equipped with a cubical C 0-system, where S is the monad of free strict cubical ∞-categories on cubical sets. Actions of this monad are on cubical sets with no...

Modèles algébriques d'(infini,n)-catégories faibles cubiques" Dans cet exposé nous allons construire pour chaque entier n, une monade Wn sur la catégorie des ensembles cubiques (appelés ensembles pré-cubiques par Richard Steiner) dont les algèbres sont des modèles d'(infini,n)-catégories faibles cubiques. Ainsi pour n=0 on obtient une monade W0...

These notes follows the articles [4, 5, 8] which show how powerful can be the method of Stretchings initiated with the Globular Geometry by Jacques Penon in [10] , to weakened strict higher structures. Here we adapt this method to weakened strict multiple ∞-categories, strict multiple (∞, m)-categories, and in particular we obtain algebraic models...

We start this article by rebuilding higher operads of weak higher transformations, and correct those in [7]. As in [7] we propose an operadic approach for weak higher n-transformations, for each n ∈ N, where such weak higher n-transformations are seen as algebras for specific contractible higher operads. The last chapter of this article asserts tha...

In this short talk we first briefly recall [4] how to build, for each integers n ≥ 0, monads Tn on the category Glob of globular sets which algebras are globular models of (∞,n)-categories, which have the virtue to be weak∞-categories of Penon and thus also to be weak ∞-categories of Batanin (see [2, 6]). On the other hand we are also briefly explain...

L'une des conjectures les plus importantes de la théorie des catégories supérieures, vue sous l'angle globulaire, est la la suivante : L'infini-catégorie faible des infini-catégories faibles existe dans le contexte globulaire. Thomason et Grothendieck espéraient la véracité de cette conjecture. Appelons B0 l'opérade de Batanin dont les algèbres...

In this article we introduce the notion of Fractal ω-operad emerging from a natural ω-operad associated to any coglobular object in the category of higher operads in Batanin's sense, which in fact is a coendomorphism ω-operads. We have in mind coglobular object of higher operads which algebras are kind of higher transformations. It follows that thi...

In this article we discuss examples of fractal ω-operads. Thus we show that there is an ω-operadic approach to explain existence of the globular set of globular sets 1 , the reflexive globular set of reflexive globular sets, the ω-magma of ω-magmas, and also the reflexive ω-magma of reflexive ω-magmas. Thus, even though the existence of the globula...

In this paper we define a sequence of monads T^(∞,n) (n ∈ N) on the category ∞-Gr of ∞-graphs. We conjecture that algebras for T^(∞,0), which are defined in a purely algebraic setting, are models of ∞-groupoids. More generally, we conjecture that T^(∞,n) -algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids...

First Algebraic and Globular Approach of Weak Higher Functors and Weak Higher Transformations. Published in 2008 (Cahier de Topologie et de Géomètrie Différentielle Catégorique).

ASPECTS OF GLOBULAR HIGHER CATEGORY THEORY - Volume 90 Issue 1 - CAMELL KACHOUR

In this note we propose an $\omega$-operadical way to prove the existence of the $\omega$-graph of the $\omega$-graphs and the reflexive $\omega$- graph of the reflexive $\omega$-graphs.

It is well known that strict $\omega$-categories, strict $\omega$-functors, strict natural $\omega$-transformations, and so on, form a strict $\omega$-category. A similar property for weak $\omega$-categories is one of the main hypotheses in higher category theory in the globular setting. In this paper we show that there is a natural globular $\ome...

In this short notes we propose a new notion of contractibility for coloured $\omega$-operad defined in the article published in Cahiers de Topologie et de G{\'e}om{\'e}trie Diff{\'e}rentielle Cat{\'e}gorique (2011), volume 4. We propose also an other way to build the monad for free contractible coloured $\omega$-operads.

In this paper we define a sequence of monads $\mathbb{T}^(\infty;n)$ $(n\in\mathbb{N})$ on $\infty$-$\mathbb{G}\text{r}$, the category of the $\infty$-graphs. We conjecture that algebras for $\mathbb{T}^(0;n)$ which are defined in a purely algebraic setting, are models of weak $\infty$-groupoids. And for all $n>1$ we conjecture that algebras for $\...

An overcategory with base category C is merely any functor into C. In this paper we extend the work of Dominique Bourn and Jacques Penon ("Cat\'egorification de structures d\'efinies par monade cart\'esienne") on overcategories. In particular we show that Freyd's adjoint theorem, a theorem of Barr and Wells ("Toposes, Triples and Theories"), all re...

Clemens Berger showed that Weak Omega Categories of Michael Batanin can be defined as model of a certain kind of theories that he called "homogeneous theories". By using the work of Mark Weber on the Abstract Nerves for the specific case of the $n$-Transformations ($n\in {\mathbb{N}^*}$), we show that we can also define Weak Omega Functors, Weak Om...

In [K. Kachour. D\'efinition alg\'ebrique des cellules non-strictes. Cahiers de Topologie et de G\'eom\'etrie Diff\'erentielle Cat\'egorique, 1:1-68, 2008] we pursue Penon's work in higher dimensional categories by defining non-strict infinity-functors, non-strict natural infinity-transformations, and so on, all that with Penon's frameworks i.e wit...

In this article we build globular higher operads for weak infini-functors, but also for weak infini-natural transformations, and so on.