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 tA,B : A ⊗ TB → T(A ⊗ B), called (tensorial) strength, such that the diagrams
 * [[Image:Strong monad left unit.svg]], [[Image:Strong monad associative.svg]],
 * [[Image:Strong monad unit.svg]], and [[Image:Strong monad multiplication.svg]]

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
 * [[Image:Strong monad commutation.svg]]

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, and the conversion between one and the other presentation is bijective.
 * 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$$t_{A,B}=m_{A,B}\circ(\eta_A\otimes 1_{TB}):A\otimes TB\to T(A\otimes B)$$