Hopf Categories

Abstract

We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.2010 Mathematics Subject Classification. 16T05.

Cohomology of Modules Over -categories and Co--categories

“…(3) Since M od-(C#H) is a Grothendieck category, it has enough injectives. The result is now clear from (2).…”

“…Hopf comodule categories were also studied in [24], where the authors introduced cleft H-comodule categories and extended classical results on cleft comodule algebras. More recently, Batista, Caenepeel and Vercruysse have shown in [2] that several deep theorems on Hopf modules can be extended to a categorification of Hopf algebras (see also [5]). In this paper, we will construct a Grothendieck spectral sequence that computes the higher derived Hom functors for H-equivariant modules over an H-category C. We will also construct a spectral sequence that gives the higher derived Hom functors for relative (D, H)-modules, where D is a co-H-category.…”

Descent and Galois theory for Hopf categories

“…Thus we can regard a klinear category as a multi-object version of a k-algebra. This philosophy was further examined in [3], leading to multi-object versions of bialgebras and Hopf algebras, respectively termed k-linear semi-Hopf categories and k-linear Hopf categories. It turns out that several classical properties of Hopf algebras can be generalized to Hopf categories, see [3] for some examples.…”

“…This philosophy was further examined in [3], leading to multi-object versions of bialgebras and Hopf algebras, respectively termed k-linear semi-Hopf categories and k-linear Hopf categories. It turns out that several classical properties of Hopf algebras can be generalized to Hopf categories, see [3] for some examples. One of the results in [3] is the fundamental theorem for Hopf modules, opening the way to Hopf-Galois theory.…”

“…It turns out that several classical properties of Hopf algebras can be generalized to Hopf categories, see [3] for some examples. One of the results in [3] is the fundamental theorem for Hopf modules, opening the way to Hopf-Galois theory. The main aim of this paper is to develop Hopf-Galois theory for Hopf categories.…”