Categories graded by group homomorphisms

TL;DR

This paper generalizes the concept of $ au$-graded categories by incorporating group homomorphisms, providing new structural insights and alternative descriptions for these categories.

Contribution

It introduces a generalization of $ au$-graded categories using group homomorphisms, along with a Yoneda lemma, a structure theorem, and an alternative categorical perspective.

Findings

Established a 'half-enriched' Yoneda lemma for $ au$-graded categories.

Proved a structure theorem for semisimple $ au$-graded categories.

Provided an alternative description of $ au$-graded categories via pseudofunctors.

Abstract

We generalise to a group homomorphism $\tau$ the $\chi$-graded categories of S\"{o}zer and Virelizier. These are categories in which both morphisms and objects have compatible degrees. We give a 'half-enriched' Yoneda lemma, a structure theorem for semisimple $\tau$-graded categories, and an alternative picture of $\tau$-graded categories in terms of pseudofunctors into $\mathbf{Cat}$.