arXiv subjects

Léonard Guetta

Publications and source records attributed to Léonard Guetta.

9 recordsLinked to original sources

arXiv2026

Groupoidal polygraphic homology

We show that for a 1-category C, the (ω, k)-polygraphic homology of C for any k {\geq} 1, that is taken with cofibrant resolutions in strict (ω, k)- categories, does not depend on k and is canonically isomorphic to the homology of the classifying space of C. When C is a groupoid, we also show this for k = 0. In particular, this means that the classical homology of groups can be obtained by taking cofibrant resolutions in strict ω-groupoids. In order to show these results, we develop the theory of discrete Conduché fibrations in the category of strict (ω, k)-categories, building on previous work by the first-named author.

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arXiv2026

Fibrantly-transferred model structures

We develop new techniques for constructing model structures from a given class of cofibrations, together with a class of fibrant objects and a choice of weak equivalences between them. As a special case, we obtain a more flexible version of the classical right-transfer theorem in the presence of an adjunction. Namely, instead of lifting the classes of fibrations and weak equivalences through the right adjoint, we now only do so between fibrant objects, which allows for a wider class of applications.

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arXiv2026

Lax functorialities of the comma construction for $ω$-categories

Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict $ω$-categories. To state the most general functorialities, we use the language of Gray $ω$-categories, that is, categories enriched in the category of strict $ω$-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict $ω$-categories defines a Gray $ω$-functor, that is, a morphism of Gray $ω$-categories. To makes sense of this statement, we prove that slices of Gray $ω$-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict $ω$-categories defines a Gray $ω$-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray $ω$-functors, which we plan to investigate in future work.

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arXiv2024

Double categorical model of $(\infty,1)$-categories

Building on work by Fiore-Pronk-Paoli, we construct four model structures on the category of double categories, each modeling one of the following: simplicial spaces, Segal spaces, $(\infty,1)$-categories, and $\infty$-groupoids. Additionally, we provide an explicit formula for computing homotopy colimits in these models using the Grothendieck construction. We expect the model of double categories for $(\infty,1)$-categories to play a similar role than that of the model of categories for spaces or $\infty$-groupoids in Grothendieck's study of the homotopy theory of spaces.

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arXiv2024

Homologie polygraphique des systèmes locaux

In this article, we introduce a notion of polygraphic homology of a strict $ω$-category with coefficients in a local system, generalizing the polygraphic homology with coefficients in $\mathbb Z$, introduced by François Métayer. We show that the homology of a simplicial set with coefficients in a local system coincides with the polygraphic homology of its image by the left adjoint of the Street nerve with coefficients in the corresponding local system. We define in this framework a comparison morphism between the polygraphic homology of a strict $ω$-category and the homology of its Street nerve, and we show that this morphism is an isomorphism for (1-)categories. This is not true for an arbitrary $ω$-category. Nevertheless, we conjecture that for an analogous construction in the framework of weak $ω$-categories ``à la Grothendieck'' we would always obtain an isomorphism.

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arXiv2023

Presheaves of groupoids as models for homotopy types

We introduce the notion of groupoidal (weak) test category, which is a small category A such that the groupoid-valued presheaves over A models homotopy types in a "canonical and nice" way. The definition does not require a priori that A is a (weak) test category, but we prove twon important comparison results: (1) every weak test category is a groupoidal weak test category, (2) a category is a test category if and only if it is a groupoidal test category. As an application, we obtain new models for homotopy types, such as the category of groupoids internal to cubical sets with or without connections, the category of groupoids internal to cellular sets, the category of groupoids internal to semi-simplicial sets, etc. We also prove, as a by-product result, that the category of groupoids internal to the category of small categories models homotopy types.

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arXiv2021

Homology of strict $ω$-categories

In this dissertation, we compare the "classical" homology of an $ω$-category (defined as the homology of its Street nerve) with its polygraphic homology. More precisely, we prove that both homologies generally do not coincide and call homologically coherent the particular strict $ω$-categories for which polygraphic homology and homology of the nerve do coincide. The goal pursued is to find abstract and concrete criteria to detect homologically coherent $ω$-categories. For example, we prove that all (small) categories, considered as strict $ω$-categories with unit cells above dimension 1, are homologically coherent. We also introduce the notion of bubble-free 2-category and conjecture that a cofibrant 2-category is homologically coherent if and only if it is bubble-free. We also prove important results concerning free strict $ω$-categories on polygraphs (also known as computads), such as the fact that if F is a discrete Conduché $ω$-functor from C to D and if D is a free strict $ω$-category on a polygraph, then so is C. Overall, this thesis achieves to build a general framework in which to study the homology of strict $ω$-categories using tools of abstract homotopical algebra such as Quillen's theory of model categories or Grothendieck's theory of derivators.

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arXiv2021

Homology of categories via polygraphic resolutions

In this paper, we extend a result of Lafont and M{é}tayer and prove that the polygraphic homology of a small category, defined in terms of polygraphic resolutions in the category $ω$Cat of strict $ω$-categories, is naturally isomorphic to the homology of its nerve. Along the way, we prove some results on homotopy colimits with respect to the Folk model structure and deduce a theorem which formally resembles Quillen's Theorem A.

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arXiv2020

Polygraphs and Discrete Conduch{é} $ω$-Functors

We define a class of morphisms between strict $ω$-categories called discrete Conduch{é} $ω$-functors that generalize discrete Conduch{é} functors between 1-categories and we study their properties related to polygraphs. The main result we prove is that for every discrete Conduch{é} $ω$-functor, if its target is a free strict $ω$-category on a polygraph then so is its source.

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