Searcharxiv⌕ Search

arXiv subjects

François Métayer

Publications and source records attributed to François Métayer.

5 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.

math.CT↗
arXiv2025

Polygraphs: From Rewriting to Higher Categories

Polygraphs are a higher-dimensional generalization of the notion of directed graph. Based on those as unifying concept, this monograph on polygraphs revisits the theory of rewriting in the context of strict higher categories, adopting the abstract point of view offered by homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. It is meant to be progressive, with little requirements on the background of the reader, apart from basic category theory, and is illustrated with algorithmic computations on algebraic structures. The second half introduces and studies the general notion of n-polygraph, dealing with the homotopy theory of those. It constructs the folk model structure on the category of strict higher categories and exhibits polygraphs as cofibrant objects. This allows extending to higher dimensional structures the coherence results developed in the first half.

math.CT↗
arXiv2023

Orientals as free algebras

The aim of this paper is to give an alternative construction of Street's cosimplicial object of orientals, based on an idea of Burroni that orientals are free algebras for some algebraic structure on strict $ω$-categories. More precisely, following Burroni, we define the notion of an expansion on an $ω$-category and we show that the forgetful functor from strict $ω$-categories endowed with an expansion to strict $ω$-categories is monadic. By iterating this monad starting from the empty $ω$-category, we get a cosimplicial object in strict $ω$-categories. Our main contribution is to show that this cosimplicial object is the cosimplicial objects of orientals. To do so, we prove, using Steiner's theory of augmented directed chain complexes, a general result for comparing polygraphs having same generators and same linearized sources and targets.

math.CT↗
arXiv2011

The Brown-Golasinski model structure on strict $\infty$-groupoids revisited

We prove that the folk model structure on strict $\infty$-categories transfers to the category of strict $\infty$-groupoids (and more generally to the category of strict $(\infty, n)$-categories), and that the resulting model structure on strict $\infty$-groupoids coincides with the one defined by Brown and Golasinski via crossed complexes.

math.CT↗