The Intrinsic Fundamental Group of a Linear Category

TL;DR

This paper introduces an intrinsic way to define the fundamental group of a linear category over a ring, linking it to Galois coverings and Hochschild-Mitchell cohomology, and establishing isomorphisms with Galois groups.

Contribution

It provides a new intrinsic definition of the fundamental group for linear categories and relates it to Galois coverings and Hochschild-Mitchell cohomology.

Findings

The fundamental group is the automorphism group of the fibre functor on Galois coverings.

If the universal covering exists, the fundamental group is isomorphic to its Galois group.

A canonical monomorphism relates the automorphism group to Hochschild-Mitchell cohomology.

Abstract

We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.