Homologie polygraphique des syst\`emes locaux

TL;DR

This paper introduces a generalized polygraphic homology for strict ω-categories with local coefficients, compares it with simplicial set homology, and establishes isomorphisms in specific cases, proposing conjectures for weak ω-categories.

Contribution

It extends polygraphic homology to local coefficients, compares it with simplicial homology, and proves isomorphisms for strict 1-categories, suggesting broader applicability.

Findings

Polygraphic homology with local coefficients generalizes previous notions.

Homology of simplicial sets with local coefficients matches polygraphic homology of their ω-category images.

Isomorphism holds for strict 1-categories but not for arbitrary ω-categories.

Abstract

In this article, we introduce a notion of polygraphic homology of a strict $\omega$-category with coefficients in a local system, generalizing the polygraphic homology with coefficients in $\mathbb Z$, introduced by Fran\c{c}ois M\'etayer. 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 $\omega$-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 $\omega$-category. Nevertheless, we conjecture that for an analogous construction in the framework of weak $\omega$-categories ``\`a la Grothendieck'' we would always obtain an isomorphism.