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 [1]).

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 $B$. [2]

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.

## References

1. ^ Moggi, Eugenio (July 1991). "Notions of computation and monads" (PDF). Information and Computation 93 (1): 55–92. doi:10.1016/0890-5401(91)90052-4.
2. ^ (ed.), Anca Muscholl (2014). Foundations of software science and computation structures : 17th (Aufl. 2014 ed.). [S.l.]: Springer. pp. 426–440. ISBN 978-3-642-54829-1.