Tannaka duality for Maschkean categories

Paddy McCrudden – 16 September 1998

Tannaka duality is concerned with the reconstruction of a coalgebra from its category of of finite dimensional representations equipped with the forgetful functor into the category of vector spaces. In this seminar I provide sufficient conditions on a braided monoidal category V such that this theorem remains true when we replace the category of vector spaces in Tannaka duality with V. I then construct a broad class of braided monoidal categories that satisfy these conditions. In a little more detail, define a Maschkean category to be a cocomplete braided monoidal abelian category such that monics split. Let V be a Maschkean category and C a coalgebra in V. I prove that C can be reconstrcuted from the functor Comod_f(C) --> C. This result is not new, and may in fact be deduced from [par96]. The problem, however is to construct a class of Maschkean categories. I prove that if V is a Maschkean categories, and H is a cobraided hopf algebra with an integral in V, then comod(H) is again Maschkean. The proof of this result may be found in my thesis (forthcoming) which will be available on my webpage. [PAR96] Bodo Pareigis, Reconstruction of Hidden Symmetries, J. Alg., 183 (1996) pages 90 - 154.