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

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

A strong monad T is said to be commutative when the diagram

commutes for all objects ${\displaystyle A}$ and ${\displaystyle B}$.[2]

One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,

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

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. ^ Muscholl, Anca, ed. (2014). Foundations of software science and computation structures : 17th (Aufl. 2014 ed.). [S.l.]: Springer. pp. 426–440. ISBN 978-3-642-54829-1.