Cubical categories for homotopy and rewriting

Maxime Lucas 1, 2
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : Higher dimensional rewriting theory was born following work by Squier on the word problem in the 80s. The goal of this work is to extend and modernize those results from Squier. In the first part of this work we show how to use rewriting to prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations, extending techniques already applied to monoidal categories. The goal of the second half of this work is to express rewriting theory in a more fitting framework. Our first step towards this goal is to define the notion of cubical (omega,p)-category, and to prove its equivalence with its globular counterpart. These structures are put to use in the final part of this thesis, in order to extend existing results of construction of polygraphic resolutions. The resolutions we consider are slightly different from the ones found in the literature. In particular the Gray tensor product of omega-categories is a key ingredient to our framework. This choice (made necessary for combinatorial reasons) is justified by homotopical considerations.
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Maxime Lucas. Cubical categories for homotopy and rewriting. Algebraic Topology [math.AT]. Université Paris 7, Sorbonne Paris Cité, 2017. English. ⟨tel-01668359⟩

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