Categories of Representations of Coalgebroids

Abstract

Fundamental results of Tannaka duality include the reconstruction of a coalgebra in the category of vector spaces from its category of representations equipped with the forgetful functor, and the characterization of those categories equipped with a functor into the category of vector spaces which are equivalent to the category of representations of some coalgebra. This paper generalizes these results by replacing the category of vector spaces by an arbitrary monoidal category and replacing coalgebras by their several-object analogue. Applications include providing sufficient conditions on a small R-linear category to ensure that its category of presheaves is equivalent to the category of representations of some R-coalgebra. In particular we construct an R-coalgebra whose category of representations is equivalent to the category of chain complexes over R.