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Georges Maltsiniotis

Publications and source records attributed to Georges Maltsiniotis.

10 recordsLinked to original sources
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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arXiv2022

Comparison of the $n$-categorical nerves

Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the classes of $n$-categorical weak equivalences that they define coincide: they are the Thomason equivalences. We give two applications of this result: the first one states that a Dyer-Kan-type equivalence for Thomason equivalences is a Thomason equivalence; the second one, fundamental, is the stability of the class of Thomason equivalences under the dualities of the category of strict $n$-categories.

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arXiv2020

Join and slices for strict $\infty$-categories

The goal of this paper is to develop a theory of join and slices for strict $\infty$-categories. To any pair of strict $\infty$-categories, we associate a third one that we call their join. This operation is compatible with the usual join of categories up to truncation. We show that the join defines a monoidal category structure on the category of strict $\infty$-categories and that it respects connected inductive limits in each variable. In particular, we obtain the existence of some right adjoints; these adjoints define $\infty$-categorical slices, in a generalized sense. We state some conjectures about the functoriality of the join and the slices with respect to higher lax and oplax transformations and we prove some first results in this direction. These results are used in another paper to establish a Quillen Theorem A for strict $\infty$-categories. Finally, in an appendix, we revisit the Gray tensor product of strict $\infty$-categories. One of the main tools used in this paper is Steiner's theory of augmented directed complexes.

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arXiv2020

A Quillen's Theorem A for strict $\infty$-categories II: the $\infty$-categorical proof

This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the current paper, we give a conceptual proof of an $\infty$-categorical nature of the same theorem. This proof is based on the theory of join and slices for strict $\infty$-categories developed by the authors in a previous paper, and on a comma construction for strict $\infty$-categories generalizing classical comma categories and Gray's comma 2-categories. This $\infty$-categorical comma construction is used by the first author in another paper to prove a generalization of Quillen's Theorem B to strict $\infty$-categories. We believe that the importance of this comma construction in the theory of $\infty$-categories goes far beyond the scope of homotopy theory.

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arXiv2018

A Quillen's Theorem A for strict $\infty$-categories I: the simplicial proof

The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely $\infty$-categorical methods.

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arXiv2015

The homotopy type of the $\infty$-category associated to a simplicial complex

This paper is part of a series of papers about homotopy theory of strict $n$-categories. In the first paper of this series, we gave conditions that guarantee the existence of a Thomason model category structure on the category of strict $n$-categories. The main goal of our paper is to show one of these conditions. To do so, we associate to any simplicial complex a strict $\infty$-category generated by a computad. We conjecture that this $\infty$-category has the same homotopy type as the corresponding simplicial complex and we prove this conjecture when the simplicial complex comes from a poset. We introduce the notion of a quasi-initial object of an $\infty$-category and we show that Street's orientals admit such an object. One of the main tools used in this text is Steiner's theory of augmented directed complexes.

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arXiv2014

Towards a Thomason model structure on the category of strict n-categories

The purpose of this article is to present ideas towards obtaining a model category structure on the category of small strict n-categories, generalizing the one obtained by Thomason on ordinary categories. Following ideas of Grothendieck and Cisinski, we obtain an "abstract Thomason theorem", which easily implies the classical Thomason theorem. We deduce a 2-categorical Thomason theorem, an incorrect proof of which has been published by K. Worytkiewicz, K. Hess, P. Parent and A. Tonks. For n > 2, we isolate sufficient conditions to obtain an n-categorical Thomason theorem. These conditions will be investigated in further work.

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arXiv2011

Homotopical exact squares and derivators

The aim of this paper is to generalize in a homotopical framework the notion of exact square introduced by René Guitart, and explain the relationship between this generalization and the theory of derivators.

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arXiv2006

Quillen's adjunction theorem for derived functors, revisited

The aim of this paper is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for "absolute" derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used.

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