Semisimple Hopf monoidal categories are group theoretical

The celebrated Tannaka-Krein reconstruction theorem [Del07] provides an equivalence between well-behaved monoidal categories equipped with a functor to the category of vector spaces, and the category of bialgebras. This equivalence, itself being a monoidal functor, can be used to transport various structures and properties between the two contexts.

Motivated by both physical and purely mathematical considerations, the Tannaka-Krein reconstruction theorem was expected to categorify as early as 1997 (see [Neu97]) and proven to do so for multifusion Hopf monoidal categories in[Gre23]. Here, multifusion Hopf monoidal categories are the categorical analogue of the finite dimensional semisimple Hopf algebras.

The surprise is that Hopf monoidal structures on multifusion categories admit a simple classification (categorifying the results of Natale [Nat03]) in terms of exact factorizations of finite groups and group cohomology, in marked contrast to the decategorified situation.

This classification is obtained as a synthesis of results of Thibault Deccopet and Matthew Yu [DY25], and the previous work [Gre23]. Moreover, a connection with the Kac exact sequence [Kac68] provides a reasonable ansatz about the classification of semisimple Hopf monoidal $n$-categories.

In the talk, we will introduce a definition of Hopf monoidal categories purely in terms of monoidal 1-categories, and provide some motivation for these categorical structures as well as a statement of the classification result.