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

File:Strong monad left unit.svg, File:Strong monad unit.svg,

File:Strong monad associative.svg,

and

File: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

File: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,

• 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.