Linear Representations and Frobenius Morphisms of Groupoids

Given a morphism of (small) groupoids with injective object map, we provide su cient 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 bre of the (left or right) pull-back biset has nitely many orbits. Our results extend and clarify the classical Frobenius reciprocity formulae in the theory of nite groups, and characterize Frobenius extension of algebras with enough orthogonal idempotents.Research supported by the Spanish Ministerio de Economía y Competitividad and the European Union FEDER, grant MTM2016-77033-