Strong monad

From Wikipedia, the free encyclopedia

You have new messages (last change).

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

Image:Strong_monad_left_unit.png, Image:Strong_monad_unit.png,
Image:Strong_monad_assoc.png,

and

Image:Strong_monad_mult.png

commute for every object A, B and C.

[edit] 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

Image:Strong_monad_commutation.png

commutes for every 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,η,μ,t) defines a symmetric monoidal monad (T,η,μ,m) by
m_{A,B}=\mu_{A\otimes B}\circ Tt'_{A,B}\circ Tt_{TA,B}:TA\otimes TB\to T(A\otimes B)
  • and conversely a symmetric monoidal monad (T,η,μ,m) defines a commutative strong monad (T,η,μ,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.

[edit] References

Retrieved from "http://en.wikipedia.org../../../s/t/r/Strong_monad.html"