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

[math]\displaystyle{ t'_{A,B}=T(\gamma_{B,A})\circ t_{B,A}\circ\gamma_{TA,B} : TA\otimes B\to T(A\otimes B) }[/math].

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

commutes for all objects [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math].

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

• a commutative strong monad [math]\displaystyle{ (T,\eta,\mu,t) }[/math] defines a symmetric monoidal monad [math]\displaystyle{ (T,\eta,\mu,m) }[/math] by

[math]\displaystyle{ m_{A,B}=\mu_{A\otimes B}\circ Tt'_{A,B}\circ t_{TA,B}:TA\otimes TB\to T(A\otimes B) }[/math]

• and conversely a symmetric monoidal monad [math]\displaystyle{ (T,\eta,\mu,m) }[/math] defines a commutative strong monad [math]\displaystyle{ (T,\eta,\mu,t) }[/math] by

[math]\displaystyle{ t_{A,B}=m_{A,B}\circ(\eta_A\otimes 1_{TB}):A\otimes TB\to T(A\otimes B) }[/math]

and the conversion between one and the other presentation is bijective.

References

• Anders Kock (1972). "Strong functors and monoidal monads". Archiv der Mathematik 23: 113–120. doi:10.1007/BF01304852. http://home.imf.au.dk/kock/SFMM.pdf. 
• Jean Goubault-Larrecq, Slawomir Lasota and David Nowak (2005). "Logical Relations for Monadic Types". Mathematical Structures in Computer Science 18 (6): 1169. doi:10.1017/S0960129508007172.