Linear representations and Frobenius morphisms of groupoids. (English) Zbl 1409.18003
Summary: 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.
MSC:
| 18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |
| 20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |
| 16D90 | Module categories in associative algebras |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 18D35 | Structured objects in a category (MSC2010) |
Keywords:
restriction; inductions and co-induction functors; groupoids-bisets; translation groupoids; Frobenius extensions; Frobenius reciprocity formulaReferences:
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