Homology of strict $\omega$-categories

TL;DR

This paper compares classical and polygraphic homology of strict ω-categories, establishing criteria for when they coincide, and explores properties of free ω-categories and homological coherence using homotopical algebra tools.

Contribution

It introduces the concept of homological coherence for strict ω-categories and provides criteria and results for detecting and analyzing this property.

Findings

All small categories are homologically coherent as strict ω-categories.

Conjecture: a cofibrant 2-category is homologically coherent iff it is bubble-free.

If D is a free strict ω-category on a polygraph and F is a discrete Conduché ω-functor, then C is also free.

Abstract

In this dissertation, we compare the "classical" homology of an $\omega$-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 $\omega$-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 $\omega$-categories. For example, we prove that all (small) categories, considered as strict $\omega$-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 $\omega$-categories on polygraphs (also known as computads), such as the fact that if F is a discrete Conduch\'e $\omega$-functor from C to D and if D is a free strict $\omega$-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 $\omega$-categories using tools of abstract homotopical algebra such as Quillen's theory of model categories or Grothendieck's theory of derivators.