Monoidal monad explained
In category theory, a branch of mathematics, a monoidal monad
is a
monad
on a
monoidal category
such that the functor
is a
lax monoidal functor and the natural transformations
and
are
monoidal natural transformations. In other words,
is equipped with coherence maps
and
satisfying
certain properties (again: they are lax monoidal), and the unit
and multiplication
are
monoidal natural transformations. By monoidality of
, the morphisms
and
are necessarily equal.
of monoidal categories, lax monoidal functors, and monoidal natural transformations.
Opmonoidal monads
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".
An opmonoidal monad is a monad
in the 2-category of
monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad
on a monoidal category
together with coherence maps
and
satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit
and the multiplication
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.
[3] An easy example for the monoidal category
of vector spaces is the monad
, where
is a
bialgebra. 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.
Properties
- The Kleisli category of a monoidal monad has a canonical monoidal structure, induced by the monoidal structure of the monad, and such that the free functor is strong monoidal. The canonical adjunction between
and the Kleisli category is a
monoidal adjunction with respect to this monoidal structure, this means that the 2-category
has Kleisli objects for monads.
- The 2-category of monads in
is the 2-category of monoidal monads
and it is isomorphic to the 2-category
of monoidales (or pseudomonoids) in the category of monads
, (lax) monoidal arrows between them and monoidal cells between them.
[4] - The Eilenberg-Moore category of an opmonoidal monad has a canonical monoidal structure such that the forgetful functor is strong monoidal. Thus, the 2-category
has Eilenberg-Moore objects for monads.
- The 2-category of monads in
is the 2-category of monoidal monads
and it is isomorphic to the 2-category
of monoidales (or pseudomonoids) in the category of monads
opmonoidal arrows between them and opmonoidal cells between them.
Examples
The following monads on the category of sets, with its cartesian monoidal structure, are monoidal monads:
. Indeed, there is a function
, sending a pair
(X'\subseteqX,Y'\subseteqY)
of subsets to the subset
\{(x,y)\midx\inX'andy\inY'\}\subseteqX x Y
. This function is natural in
X and
Y. Together with the unique function
as well as the fact that
are monoidal natural transformations,
is established as a monoidal monad.
The following monads on the category of sets, with its cartesian monoidal structure, are not monoidal monads
is a monoid, then
is a monad, but in general there is no reason to expect a monoidal structure on it (unless
is commutative).
Notes and References
- Moerdijk. Ieke. Monads on tensor categories. Journal of Pure and Applied Algebra. 23 March 2002. 168. 2–3. 189–208. 10.1016/S0022-4049(01)00096-2. dmy-all. free.
- Bruguières. Alain. Alexis Virelizier. Hopf monads. Advances in Mathematics. 2007. 215. 2. 679–733. 10.1016/j.aim.2007.04.011. free. dmy-all.
- McCrudden. Paddy. 2002. Opmonoidal monads. Theory and Applications of Categories. 10. 19. 469–485 . 10.1.1.13.4385.
- Zawadowski. Marek. 2011. The Formal Theory of Monoidal Monads The Kleisli and Eilenberg-Moore objects. Journal of Pure and Applied Algebra. 216. 8–9. 1932–1942. 10.1016/j.jpaa.2012.02.030. 1012.0547. 119301321 .