arXiv subjects
Publications and source records attributed to Camell Kachour.
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 $Θ^{\infty}_{\mathbb{M}^n}$ 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 $Θ^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $Θ_0$ whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict $\infty$-categories with connections. We prove that it is a cartesian monad, solving a conjecture in \cite{camark-cub-1}. This puts us in a position to describe the cubical coherator $Θ^{\infty}_W$ whose set-models are cubical weak $\infty$-categories with connections and the cubical coherator $Θ^{\infty}_{W^{0}}$ whose set-models are cubical weak $\infty$-groupoids with connections.
In this article we show how to build main aspects of our paper on globular weak $(\infty,n)$-categories, but now for the cubical geometry. Thus we define a monad on the category $\mathbb{C}\mathbb{S}ets$ of cubical sets which algebras are models of cubical weak $\infty$-categories. Also for each $n\in\mathbb{N}$ we define a monad on $\mathbb{C}\mathbb{S}ets$ which algebras are models of cubical weak $(\infty,n)$-categories. And finally we define a monad on the category $\mathbb{C}\mathbb{S}ets^2$ which algebras are models of cubical weak $\infty$-functors, and a monad on the category $\mathbb{C}\mathbb{S}ets^4$ which algebras are models of cubical weak natural $\infty$-transformations.
These notes follows the articles \cite{kamel, Cam, cam-cubique} which show how powerful can be the method of \textit{Stretchings} initiated with the \textit{Globular Geometry} by Jacques Penon in \cite{penon} , to weakened \textit{strict higher structures}. Here we adapt this method to weakened strict multiple $\infty$-categories, strict multiple $(\infty,m)$-categories, and in particular we obtain algebraic models of weak multiple $\infty$-groupoids.
In this note we propose an $ω$-operadical way to prove the existence of the $ω$-graph of the $ω$-graphs and the reflexive $ω$- graph of the reflexive $ω$-graphs.
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 ("Catégorification de structures définies par monade cartésienne") on overcategories. In particular we show that Freyd's adjoint theorem, a theorem of Barr and Wells ("Toposes, Triples and Theories"), all remain true in the context of overcategories. We also show that a free monoid construction remains valid in the context of overcategories. The motivation for this study is the development of higher categories as found in the work of Dominique Bourn and Jacques Penon ("Catégorification de structures définies par monade cartésienne").
In this short notes we propose a new notion of contractibility for coloured $ω$-operad defined in the article published in Cahiers de Topologie et de G{é}om{é}trie Diff{é}rentielle Cat{é}gorique (2011), volume 4. We propose also an other way to build the monad for free contractible coloured $ω$-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 $\mathbb{T}^(\infty;n)$ which are defined in a purely algebraic setting, are models of weak $(\infty; n)$-categories.
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 Omega Natural Transformations, and so on, as models of certain kind of colored theories which are homogeneous as well.
In [K. Kachour. Définition algébrique des cellules non-strictes. Cahiers de Topologie et de Géométrie Différentielle Catégorique, 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 with the "étirements catégoriques", where we have used an extension of this object, namely the "n-étirements catégoriques" (n belong in N). In this paper we are pursuing Batanin's work in higher dimensional categories by defining nonstrict infinity-functors, non-strict natural infinity-transformations, and so on, using Batanin's frameworks i.e with the contractible operads, where we used an extension of this object, namely the globular colored contractible operads.