symmetric monoidal (∞,1)-category of spectra
Bimonoid is a monoid and comonoid in compatible way. The compatibility is easy to formulate in symmetric monoidal categories and much harder in nonsymmetric setup.
In a symmetric monoidal category, a bimonoid (or bimonoid object) is an object $B$ equipped with a structure of a monoid and a comonoid which are compatible in one of two equivalent ways: the comultiplication and the counit are morphisms of monoids or the multiplication and the unit are morphisms of comonoids. The symmetry of the monoidal structure is involved in the definition of the tensor product $B\otimes B$ as monoids and as comonoids.
A bimonoid which additionally has an antipode (a map $s:B\to B$ satisfying an axiom that makes it act like “inverses” in a group) is called a Hopf monoid; a Hopf monoid in Vect is a Hopf algebra.
A bimonoid in Vect (with its usual tensor product) is generally called a bialgebra.
In a category with biproducts, with the biproduct as the monoidal product, every object is a bimonoid in a unique way (Café post.
More generally, in a cartesian monoidal category, every monoid object is a bimonoid in a unique way, with comultiplication being the diagonal map. Dually, every comonoid object in a cocartesian monoidal category is a bimonoid in a unique way.
If $B$ is a bimonoid in $\mathcal{C}$, then the category $Mod_B$ of $B$-modules inherits a monoidal structure such that the forgetful functor $Mod_B \to \mathcal{C}$ (the “fiber functor”) is a strong monoidal functor. For $B$-modules $M$ and $N$, we equip the tensor product $M\otimes N$ in $\mathcal{C}$ with the $B$-action given by
where $\Delta$ denotes the comultiplication of $B$ as a comonoid. The bimonoid compatibility axioms are exactly what is needed to make this a $B$-module structure, and coassociativity makes it associative. Similarly, we use the counit $B\to I$ of $B$ to give the unit object $I$ a $B$-module structure.
If $B$ moreover a Hopf monoid, then $Mod_B$ also inherits a closed monoidal structure if $\mathcal{C}$ has one; see at Hopf monoid.
These relations are known as Tannaka duality for monoids/algebras, see at structure on algebras and their module categories - table.
It is interesting how to generalize this notion in various nonsymmetric situations, for example involving braidings, or more generally in duoidal categories, or in relative situations over noncommutative rings (e.g. Takeuchi bialgebroids). In the completely noncommutative situation of the monoidal category of endofunctors, one can look at various compatibilities between monads and comonads or monads and tensor products, for example involving distributive laws.
As far as compatibility with tensor product is concerned, there is a notion of a bimonad and involves a version of distributive laws, hence it is related to a lifting problem:
Ieke Moerdijk, Monads on tensor categories, Category theory 1999 (Coimbra), J. Pure Appl. Algebra 168 (2002), no. 2-3, 189–208 (MR2003e:18012)
Kornél Szlachányi, The monoidal Eilenberg–Moore construction and bialgebroids, J. Pure Appl. Algebra 182, no. 2–3 (2003) 287–315; Adjointable monoidal functors and quantum groupoids, math.QA/0301253
Szlachányi uses earlier analysis of
P. Schauenburg, Bialgebras over noncommutative rings, and a structure theorem for Hopf bimodules, Applied Categorical Structures 6, 193-222 (1998) doi
H.E. Porst, On categories of monoids, comonoids and bimonoids, Quaest. Math. 31 (2008) 127-139 MR2010d:18010 doi
For a dual version see
There is also a more subtle notion also called Hopf monad in
In some of these generalized cases, one does not have a good notion of of antipode, so that the difference between bimonoids and Hopf monoids has to be stated in different terms. A similar case is in the case of Hopf algebroids over a noncommutative base:
Brian Day, Ross Street, Monoidal bicategories and Hopf algebroids, Advances in Math. 129, 1 (1997) 99–157
Gabi Böhm, Hopf algebroids, (a chapter of) Handbook of algebra, arxiv:math.RA/0805.3806; An alternative notion of Hopf algebroid; in “Hopf algebras in noncommutative geometry and physics”, 31–53, Lec. Notes in Pure and Appl. Math. 239, Dekker, New York 2005; math.QA/0301169