On inverse categories and transfer in cohomology

Abstract

AbstractIt follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories $\mathcal{C}$, being a Mackey functor on $\mathcal{C}$ is equivalent to being extendible to a suitable inverse category containing $\mathcal{C}$. We further show that extensions of inverse categories by abelian groups are again inverse categories.