Categories graded by group homomorphisms

Abstract

We generalise to a group homomorphism $\ddot{I}„$ the $\ddot{I}‡$-graded categories of Sözer 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 $\ddot{I}„$-graded categories, and an alternative picture of $\ddot{I}„$-graded categories in terms of pseudofunctors into $\mathbf{Cat}$.