Duoidal categories (under the original name `2-monoidal category') were introduced by Aguiar and Mahajan. These are pseudomonoids in the Cartesian monoidal 2-category of monoidal categories – (op)lax monoidal functors – (op)lax monoidal natural transformations. Hence they are categories with two related monoidal structures. A bimonoid in a duoidal category is an object of the category carrying the structures of a monoid with respect to the first monoidal product and a comonoid with respect to the second monoidal product. The compatibility conditions are formulated in terms of the coherence morphisms relating both monoidal structures.
In this talk we shall compare the various principles that can be used to define a bimonoid with some additional `Hopf' property.
The talk is based on joint results with Yuanyuan Chen and Liangyun Zhang and a most recent work in progress with Steve Lack.