In category theory, a strong monad over a monoidal category $({\mathcal C},\otimes,I)$ is a monad $(T,\eta,\mu)$ together with a natural transformation $t_{A,B} : A\otimes TB\to T(A\otimes B)$, 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

$t'_{A,B}=T(\gamma_{B,A})\circ t_{B,A}\circ\gamma_{TA,B} : TA\otimes B\to T(A\otimes B)$.

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
$m_{A,B}=\mu_{A\otimes B}\circ Tt'_{A,B}\circ t_{TA,B}:TA\otimes TB\to T(A\otimes B)$
• and conversely a symmetric monoidal monad $(T,\eta,\mu,m)$ defines a commutative strong monad $(T,\eta,\mu,t)$ by
$t_{A,B}=m_{A,B}\circ(\eta_A\otimes 1_{TB}):A\otimes TB\to T(A\otimes B)$

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