On universal gradings, versal gradings and Schurian generated categories

TL;DR

This paper explores the concept of universal and versal gradings in categories over a field, establishing the fundamental grading group as isomorphic to the Grothendieck fundamental group and proving the existence of universal gradings for Schurian generated categories.

Contribution

It introduces the fundamental grading group for categories over a field and proves its isomorphism to the Grothendieck fundamental group, also establishing the existence of universal gradings in Schurian generated categories.

Findings

Fundamental grading group is isomorphic to Grothendieck fundamental group.

Universal grading exists for Schurian generated categories.

Examples include categories with and without universal or versal gradings.

Abstract

Categories over a field $k$ can be graded by different groups in a connected way; we consider morphisms between these gradings in order to define the fundamental grading group. We prove that this group is isomorphic to the fundamental group \`a la Grothendieck as considered in previous papers. In case the $k$-category is Schurian generated we prove that a universal grading exists. Examples of non Schurian generated categories with universal grading, versal grading or none of them are considered.