Categories graded by group homomorphisms
Jonathan Davies
TL;DR
The paper generalises χ-graded categories to τ-graded categories via a group homomorphism $\tau: H\to G$, introducing a half-enriched Yoneda lemma and shift objects to manage the richer degree structure. It develops a robust framework for semisimplicity in the τ-graded setting, including a concrete structure theorem that classifies semisimple τ-graded categories as finite direct sums of indecomposable blocks $\mathcal{M}_{\tau}(L,\psi)\langle g\rangle$. A key contribution is the detailed link between τ-graded categories and $H$-module categories, formalised through 2-equivalences $\Cat_{\tau}^{\shifts} \simeq \mathrm{ModCat}[H]_{\tau}$ and related functors, which recasts graded categorical data in module-categorical terms via shifts and pseudofunctors into $\mathbf{Cat}$. These results connect homotopy-theoretic motivations from HQFTs to explicit, computable classifications and equivalences, enabling a unified treatment of graded categories, shifts, and module-category structures with potential applications to crossed-module frameworks and beyond.
Abstract
We generalise to a group homomorphism $τ$ the $χ$-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 $τ$-graded categories, and an alternative picture of $τ$-graded categories in terms of pseudofunctors into $\mathbf{Cat}$.