Definitions

In category theory, a strong monad over a monoidal category $\left(C,otimes,I\right)$ is a monad $\left(T,eta,mu\right)$ together with a natural transformation $t_\left\{A,B\right\} : Aotimes TBto T\left(Aotimes B\right)$, called (tensorial) strength, such that the diagrams
, ,
,
and
commute for every object $A$, $B$ and $C$.

For every strong monad T on a symmetric monoidal category, a costrength natural transformation can be defined by

$t\text{'}_\left\{A,B\right\}=T\left(gamma_\left\{B,A\right\}\right)circ t_\left\{B,A\right\}circgamma_\left\{TA,B\right\} : TAotimes Bto T\left(Aotimes B\right)$.
A strong monad T is said to be commutative when the diagram
commutes for all objects $A$ and $B$.

One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,

• a commutative strong monad $\left(T,eta,mu,t\right)$ defines a symmetric monoidal monad $\left(T,eta,mu,m\right)$ by

$m_\left\{A,B\right\}=mu_\left\{Aotimes B\right\}circ Tt\text{'}_\left\{A,B\right\}circ Tt_\left\{TA,B\right\}:TAotimes TBto T\left(Aotimes B\right)$

• and conversely a symmetric monoidal monad $\left(T,eta,mu,m\right)$ defines a commutative strong monad $\left(T,eta,mu,t\right)$ by

$t_\left\{A,B\right\}=m_\left\{A,B\right\}circ\left(eta_Aotimes 1_\left\{TB\right\}\right):Aotimes TBto T\left(Aotimes B\right)$
and the conversion between one and the other presentation is bijective.

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