Strong monad

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In category theory, a strong monad over a monoidal category ({\mathcal C},\otimes,I) is a monad (T,\eta,\mu) together with a natural transformation t_{A,B} : A\otimes TB\to T(A\otimes B), called (tensorial) strength, such that the diagrams

Strong monad left unit.png, Strong monad unit.png,

Strong monad assoc.png,


Strong monad mult.png

commute for every object A, B and C.

Commutative strong monads[edit]

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

Strong monad commutation.png

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.