In category theory, a branch of mathematics, a monoidal monad
is a lax monoidal functor and the natural transformations
are monoidal natural transformations.
is equipped with coherence maps
satisfying certain properties (again: they are lax monoidal), and the unit
are monoidal natural transformations.
are necessarily equal.
All of the above can be compressed into the statement that a monoidal monad is a monad in the 2-category
of monoidal categories, lax monoidal functors, and monoidal natural transformations.
Opmonoidal monads have been studied under various names.
Ieke Moerdijk introduced them as "Hopf monads",[1] while in works of Bruguières and Virelizier they are called "bimonads", by analogy to "bialgebra",[2] reserving the term "Hopf monad" for opmonoidal monads with an antipode, in analogy to "Hopf algebras".
, η , μ )
monoidal categories, oplax monoidal functors and monoidal natural transformations.
That means a monad
together with coherence maps
satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit
into opmonoidal natural transformations.
Alternatively, an opmonoidal monad is a monad on a monoidal category such that the category of Eilenberg-Moore algebras has a monoidal structure for which the forgetful functor is strong monoidal.
[1][3] An easy example for the monoidal category
of vector spaces is the monad
[2] The multiplication and unit of
define the multiplication and unit of the monad, while the comultiplication and counit of
give rise to the opmonoidal structure.
The algebras of this monad are right
-modules, which one may tensor in the same way as their underlying vector spaces.
The following monads on the category of sets, with its cartesian monoidal structure, are monoidal monads: The following monads on the category of sets, with its cartesian monoidal structure, are not monoidal monads