Fundamental group of Schurian categories and the Hurewicz isomorphism

TL;DR

This paper establishes a connection between the fundamental group of a CW-complex associated with a Schurian k-category and the category's intrinsic fundamental group, also linking it to Hochschild-Mitchell cohomology.

Contribution

It extends previous results by constructing a CW-complex for Schurian categories and proving an isomorphism between fundamental groups and Hochschild-Mitchell cohomology.

Findings

Fundamental group of the CW-complex matches the intrinsic fundamental group.

Hurewicz morphism is an isomorphism between abelian characters and Hochschild-Mitchell cohomology.

Extension of previous results by J.C. Bustamante.

Abstract

Let k be a field. We attach a CW-complex to any Schurian k-category and we prove that the fundamental group of this CW-complex is isomorphic to the intrinsic fundamental group of the k-category. This extends previous results by J.C. Bustamante. We also prove that the Hurewicz morphism from the vector space of abelian characters of the fundamental group to the first Hochschild-Mitchell cohomology vector space of the category is an isomorphism.