Tannaka duality for Maschkean categories

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 monoïdal category image, and finite dimensional vector spaces are replaced by those objects of image with a left dual. Sufficient conditions on image are given ensuring that a coalgebra in image may be reconstructed from those representations whose underlying object of image has a left dual. When the braiding on image is a symmetry, these conditions also suffice to reconstruct certain extra structure on a comonoid in image from the corresponding extra structure on its category of representations. A broad class of categories satisfying these conditions, called Maschkean categories, are then constructed.