In category theory, a strong monad over a monoidal category $(C,otimes,I)$ is a monad $(T,eta,mu)$ together with a natural transformation $t\_\{A,B\}\; :\; Aotimes\; TBto\; T(Aotimes\; B)$, called (tensorial) strength, such that the diagrams

, ,

,

and

commute for every object $A$, $B$ and $C$.

Commutative strong monads

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

$t\text{'}\_\{A,B\}=T(gamma\_\{B,A\})circ\; t\_\{B,A\}circgamma\_\{TA,B\}\; :\; TAotimes\; Bto\; T(Aotimes\; B)$.

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 $(T,eta,mu,t)$ defines a symmetric monoidal monad $(T,eta,mu,m)$ by

$m\_\{A,B\}=mu\_\{Aotimes\; B\}circ\; Tt\text{'}\_\{A,B\}circ\; Tt\_\{TA,B\}:TAotimes\; TBto\; T(Aotimes\; B)$

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

$t\_\{A,B\}=m\_\{A,B\}circ(eta\_Aotimes\; 1\_\{TB\}):Aotimes\; TBto\; T(Aotimes\; B)$

and the conversion between one and the other presentation is bijective.

References

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• Jean Goubault-Larrecq, Slawomir Lasota and David Nowak (2005). "Logical Relations for Monadic Types".