Introduction to duoidal categories via the Eckmann-Hilton argument.

Abstract:
Duoidal categories carry two compatible monoidal structures. We will go over the basics of duoidal categories, illustrating with a number of examples. As monoidal categories provide a context for monoids, duoidal categories provide one for duoids and bimonoids. Our main goal is to discuss a number of versions of the classical Eckmann-Hilton argument which may be formulated in this setting. As an application we will obtain the commutativity of the cup product on the cohomology of a bimonoid with coefficients in a duoid, an extension of a familiar result for group and bialgebra cohomology with trivial coefficients. The lectures borrow on earlier work in collaboration with Swapneel Mahajan on the foundations of duoidal categories (2010). The main results are from ongoing work with Javier Coppola. We also rely on work of Richard Garner and Ignacio López-Franco (2016). Familiarity with the basics of monoidal categories and monoidal functors, as well as with the basics of Hochschild cohomology, would be helpful, although these notions will be reviewed.

Programa:

Lecture 1:
A. The classical Eckmann-Hilton argument
B. Monoidal categories and functors
C. The cup product

Lecture 2:
A. Duoidal categories: definition and examples
B. Bimonoids and duoids
C. Functors between duoidal categories

Lecture 3:
A. Duoids and commutativity in duoidal categories
B. Eckmann-Hilton for duoidal functors
C. The Hochschild complex as a duoidal functor