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

, ,

Error creating thumbnail: Unable to save thumbnail to destination, 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

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

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

$m_{A,B}={\mu }_{A\otimes B}\circ Tt{\text{'}}_{A,B}\circ t_{TA,B}:TA\otimes TB\to T(A\otimes B)$

• 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

• 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.