Tannaka duality for Maschkean categories

Abstract

Tannaka duality is concerned with the reconstruction of a coalgebra in the category of vector spaces from its category of finite dimensional representations equipped with its forgetful functor. It is further concerned with the reconstruction of extra structure on a coalgebra from the corresponding extra structure on its category of representations. This article provides a generalization of Tannaka duality where the category of vector spaces is replaced by an arbitrary braided monoı̈dal category V , and finite dimensional vector spaces are replaced by those objects of V with a left dual. Sufficient conditions on V are given ensuring that a coalgebra in V may be reconstructed from those representations whose underlying object of V has a left dual. When the braiding on V is a symmetry, these conditions also suffice to reconstruct certain extra structure on a comonoid in V from the corresponding extra structure on its category of representations. A broad class of categories satisfying these conditions, called Maschkean categories, are then constructed.