### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 019, 33 pages      arXiv:1806.09327      https://doi.org/10.3842/SIGMA.2019.019

### Linear Representations and Frobenius Morphisms of Groupoids

Juan Jesús Barbarán Sánchez a and Laiachi El Kaoutit b

Received June 26, 2018, in final form February 22, 2019; Published online March 12, 2019

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.

Key words: Linear representations of groupoids; restriction, inductions and co-induction functors; groupoids-bisets; translation groupoids; Frobenius extensions; Frobenius reciprocity formula.

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