Abstract :Bihom bialgebras viewed as bimonoids
In recent years, lead by diverse motivations, several different generalizations of Hopf algebra have been proposed. Their similar features naturally raise the question whether they are instances of the same, more general notion. In [2,3] several examples, such as groupoids, Hopf monoids in braided monoidal categories, Hopf algebroids over central base algebras, weak Hopf algebras, Turaev’s Hopf group algebras, Batista-Caenepeel-Vercruysse’s Hopf categories and Bruguiéres’ Hopf polyads were unified as Hopf monoids in suitable duoidal (called 2-monoidal in ) categories.
BiHom-bimonoids in  however, do not seem to fit this picture. In order to find a framework describing them, we introduce monoidal categories whose monoidal products of any posiitive number of factors are lax coherent and whose nullary products are oplax coherent. We call them Lax+Oplax0-monoidal. Dually, we consider Lax0Oplax+-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define Lax+ 0 Oplax0+ -duoidal categories with compatible Lax+Oplax0- and Lax0Oplax+-monoidal structures. We introduce comonoids in Lax+Oplax0-monoidal categories, monoids in Lax0Oplax+-monoidal categories and bimonoids in Lax+ 0Oplax0 + -duoidal categories.
The unital BiHom-monoids, counital BiHom-comonoids, and unital and counital BiHom-bimonoids of  are identified with the monoids, comonoids and bimonoids in a suitable Lax+ 0Oplax0+ -duoidal category, constructed by a method generalized from .
This is a joint work with Joost Vercruysse at ULB, Brussels.
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