Strong monad
# Strong monad

In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation $t_{A,B}$ : A ⊗ TB → T(A ⊗ B), 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.

## Commutative strong monads

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.

