Matrices, relations, and group representations

Abstract

Motivated by the search for a good notion of a “selfdual” object in a symmetric monoidal category, we give an abstract notion of commutative separable algebra in a symmetric monoidal category V , such that when V is the category of modules over a commutative ring k we obtain the usual commutative separable algebras. Developing the calculus based on such an abstract notion we are able to prove that the dual category of separable algebras in any compact closed, additive category with coequalizers is in fact a boolean pretopos. We apply this result to give a simple characterization of the categories of continuous representations of profinite groups in discrete finite dimensional k-vector spaces. Finally we show that the same calculus can be applied to symmetric monoidal categories of relations to give an essentially algebraic characterization of such categories.