{{Refimprove|date=May 2014}} In category theory, a branch of mathematics, a '''monoidal monad''' <math>(T,\eta,\mu,T_{A,B},T_0)</math> is a monad <math>(T,\eta,\mu)</math> on a monoidal category <math>(C,\otimes,I)</math> such that the functor <math>T:(C,\otimes,I)\to(C,\otimes,I)</math> is a lax monoidal functor and the natural transformations <math>\eta</math> and <math>\mu</math> are monoidal natural transformations. In other words, <math>T</math> is equipped with coherence maps <math>T_{A,B}:TA\otimes TB\to T(A\otimes B)</math> and <math>T_0:I\to TI</math> satisfying certain properties (again: they are lax monoidal), and the unit <math>\eta: id \Rightarrow T</math> and multiplication <math>\mu:T^2\Rightarrow T</math> are monoidal natural transformations. By monoidality of <math>\eta</math>, the morphisms <math>T_0</math> and <math>\eta_I</math> are necessarily equal.
All of the above can be compressed into the statement that a monoidal monad is a monad in the 2-category <math>\mathsf{MonCat}</math> 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",<ref name="moerdijk">{{cite journal|last=Moerdijk|first=Ieke|title=Monads on tensor categories|journal=Journal of Pure and Applied Algebra|date=23 March 2002|volume=168|issue=2–3|pages=189–208|doi=10.1016/S0022-4049(01)00096-2|df=dmy-all|doi-access=free}}</ref> while in works of Bruguières and Virelizier they are called "bimonads", by analogy to "bialgebra",<ref name="bruguieres">{{cite journal|last=Bruguières|first=Alain|author2=Alexis Virelizier|title=Hopf monads|journal=Advances in Mathematics|year=2007|volume=215|issue=2|pages=679–733|doi=10.1016/j.aim.2007.04.011|doi-access=free|df=dmy-all}}</ref> reserving the term "Hopf monad" for opmonoidal monads with an antipode, in analogy to "Hopf algebras".
An '''opmonoidal monad''' is a monad <math>(T,\eta,\mu)</math> ''in'' the 2-category of <math>\mathsf{OpMonCat}</math> monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad <math>(T,\eta,\mu)</math> ''on'' a monoidal category <math>(C,\otimes,I)</math> together with coherence maps <math>T^{A,B}:T(A\otimes B) \to TA\otimes TB</math> and <math>T^0:TI\to I</math> satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit <math>\eta</math> and the multiplication <math>\mu</math> 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.<ref name="moerdijk" /><ref name=":0">{{Cite journal|last=McCrudden|first=Paddy|year=2002|title=Opmonoidal monads|url=http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html|journal=Theory and Applications of Categories|volume=10|issue=19|pages=469–485 |citeseerx=10.1.1.13.4385}}</ref>
An easy example for the monoidal category <math>\operatorname{Vect}</math> of vector spaces is the monad <math>- \otimes A</math>, where <math>A</math> is a bialgebra.<ref name="bruguieres" /> The multiplication and unit of <math>A</math> define the multiplication and unit of the monad, while the comultiplication and counit of <math>A</math> give rise to the opmonoidal structure. The algebras of this monad are right <math>A</math>-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 <math>C</math> and the Kleisli category is a monoidal adjunction with respect to this monoidal structure, this means that the 2-category <math>\mathsf{MonCat}</math> has Kleisli objects for monads. * The 2-category of monads in <math>\mathsf{MonCat}</math> is the 2-category of monoidal monads <math>\mathsf{Mnd(MonCat)}</math> and it is isomorphic to the 2-category <math>\mathsf{Mon(Mnd(Cat))}</math> of monoidales (or pseudomonoids) in the category of monads <math>\mathsf{Mnd(Cat)}</math>, (lax) monoidal arrows between them and monoidal cells between them.<ref name=":1">{{Cite journal|last=Zawadowski|first=Marek|year=2011|title=The Formal Theory of Monoidal Monads The Kleisli and Eilenberg-Moore objects|journal=Journal of Pure and Applied Algebra|volume=216|issue=8–9|pages=1932–1942|doi=10.1016/j.jpaa.2012.02.030|arxiv=1012.0547|s2cid=119301321 }}</ref> * The Eilenberg-Moore category of an opmonoidal monad has a canonical monoidal structure such that the forgetful functor is strong monoidal.<ref name="moerdijk" /> Thus, the 2-category <math>\mathsf{OpmonCat}</math> has Eilenberg-Moore objects for monads.<ref name=":0" /> * The 2-category of monads in <math>\mathsf{OpmonCat}</math> is the 2-category of monoidal monads <math>\mathsf{Mnd(OpmonCat)}</math> and it is isomorphic to the 2-category <math>\mathsf{Opmon(Mnd(Cat))}</math> of monoidales (or pseudomonoids) in the category of monads <math>\mathsf{Mnd(Cat)}</math> opmonoidal arrows between them and opmonoidal cells between them.<ref name=":1" />
==Examples== The following monads on the category of sets, with its cartesian monoidal structure, are monoidal monads: * The power set monad <math>(\mathbb{P},\varnothing,\cup)</math>. Indeed, there is a function <math>\mathbb{P}(X)\times\mathbb{P}(Y)\to\mathbb{P}(X\times Y)</math>, sending a pair <math>(X'\subseteq X,Y'\subseteq Y)</math> of subsets to the subset <math>\{(x,y)\mid x\in X'\text{ and } y\in Y'\}\subseteq X\times Y</math>. This function is natural in ''X'' and ''Y''. Together with the unique function <math>\{1\}\to\mathbb{P}(\varnothing)</math> as well as the fact that <math>\mu,\eta</math> are monoidal natural transformations, <math>\mathbb{P}</math> 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 <math>M</math> is a monoid, then <math>X\mapsto X\times M</math> is a monad, but in general there is no reason to expect a monoidal structure on it (unless <math>M</math> is commutative).
==References== {{reflist}}
{{DEFAULTSORT:Monoidal Monad}} Category:Monoidal categories