Homologie polygraphique des systèmes locaux

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 François Métayer. 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 ``à la Grothendieck'' we would always obtain an isomorphism.

56 pages, in French language. Minor corrections

References in corpus (1)

• Comparison of the $n$-categorical nerves