In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation tA,B : ATBT(AB), called (tensorial) strength, such that the diagrams , , , and commute for every object A, B and C (see Definition 3.2 in ).

If the monoidal category (C, ⊗, I) is closed then a strong monad is the same thing as a C-enriched monad.

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.