Monoidal monad explained

In category theory, a branch of mathematics, a monoidal monad

(T, η ,\mu,T _{A,B} ,T _{0)} is a monad (T, η ,\mu) on a monoidal category (C, ⊗ ,I) such that the functor T:(C, ⊗ ,I)\to(C, ⊗ ,I) is a lax monoidal functor and the natural transformations η and \mu are monoidal natural transformations. In other words, T is equipped with coherence maps T _{A,B} :TA ⊗ TB\to T(A ⊗ B) and T _{0:I\to} TI satisfying certain properties (again: they are lax monoidal), and the unit η : id ⇒ T and multiplication \mu:T ^{2 ⇒} T are monoidal natural transformations. By monoidality of η , the morphisms T _{0} and η _{I} are necessarily equal.MonCat 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

(T, η ,\mu) in the 2-category of OpMonCat monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad (T, η ,\mu) on a monoidal category (C, ⊗ ,I) together with coherence maps T ^{A,B} :T(A ⊗ B) \to TA ⊗ TB and T ^{0:TI\to} I satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit η and the multiplication \mu 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

\operatorname{Vect} of vector spaces is the monad - ⊗ A , where A is a bialgebra. The multiplication and unit of A define the multiplication and unit of the monad, while the comultiplication and counit of A give rise to the opmonoidal structure. The algebras of this monad are right A -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

C and the Kleisli category is a monoidal adjunction with respect to this monoidal structure, this means that the 2-category MonCat has Kleisli objects for monads.

• The 2-category of monads in

MonCat is the 2-category of monoidal monads Mnd(MonCat) and it is isomorphic to the 2-category Mon(Mnd(Cat)) of monoidales (or pseudomonoids) in the category of monads Mnd(Cat) , (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

OpmonCat has Eilenberg-Moore objects for monads.

• The 2-category of monads in

OpmonCat is the 2-category of monoidal monads Mnd(OpmonCat) and it is isomorphic to the 2-category Opmon(Mnd(Cat)) of monoidales (or pseudomonoids) in the category of monads Mnd(Cat) 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:

• The power set monad

( P ,\varnothing,\cup) . Indeed, there is a function P (X) x P (Y)\to P (X x Y) , sending a pair (X'\subseteq X,Y'\subseteq Y) of subsets to the subset \{(x,y)\mid x\in X' and y\in Y'\}\subseteq X x Y . This function is natural in X and Y. Together with the unique function \{ 1 \}\to P (\varnothing) as well as the fact that \mu, η are monoidal natural transformations, ( P is established as a monoidal monad.

• The probability distribution (Giry) monad.

The following monads on the category of sets, with its cartesian monoidal structure, are not monoidal monads

• If

M is a monoid, then X\mapsto X x M is a monad, but in general there is no reason to expect a monoidal structure on it (unless M is commutative).

Notes and References

1. 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.
2. Bruguières. Alain. Alexis Virelizier. Hopf monads. Advances in Mathematics. 2007. 215. 2. 679–733. 10.1016/j.aim.2007.04.011. free. dmy-all.
3. McCrudden. Paddy. 2002. Opmonoidal monads. Theory and Applications of Categories. 10. 19. 469–485 . 10.1.1.13.4385.
4. 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 .