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Books Year : 2023

Polygraphs: From Rewriting to Higher Categories

Abstract

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.

Dates and versions

hal-04322821 , version 1 (05-12-2023)

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Cite

Dimitri Ara, Albert Burroni, Yves Guiraud, Philippe Malbos, François Métayer, et al.. Polygraphs: From Rewriting to Higher Categories. In press, London Mathematical Society Lecture Note Series. ⟨hal-04322821⟩
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