Linear representations and Frobenius morphisms of groupoids

Juan Jesús Barbarán Sánchez, Laiachi EL Kaoutit

Published 2018-06-25Version 1

Given a morphism of (small) groupoids, 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 \emph{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.