Linear Representations and Frobenius Morphisms of Groupoids

TL;DR

This paper characterizes Frobenius morphisms of groupoids, extending classical Frobenius reciprocity, by establishing conditions under which induction and co-induction functors are naturally isomorphic, with implications for algebra extensions.

Contribution

It introduces the concept of Frobenius morphisms of groupoids and provides necessary and sufficient conditions for their characterization, extending classical reciprocity results.

Findings

Frobenius morphisms are characterized by isomorphic induction and co-induction functors.

Extension by a subgroupoid is Frobenius iff each fiber has finitely many orbits.

Results extend classical Frobenius reciprocity to groupoid and algebra contexts.

Abstract

Given a morphism of (small) groupoids with injective object map, we provide sufficient and necessary conditions under which the induction and co-induction functors between the categories of linear representations are naturally isomorphic. A morphism with this property is termed a Frobenius morphism of groupoids. As a consequence, an extension by a subgroupoid is Frobenius if and only if each fibre of the (left or right) pull-back biset has finitely many orbits. Our results extend and clarify the classical Frobenius reciprocity formulae in the theory of finite groups, and characterize Frobenius extension of algebras with enough orthogonal idempotents.