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 [1]).

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

commutes for all objects $A$ and $B$ .[2]

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.

gollark: Yes, but it's entirely different.

gollark: > Oh IFcoltransG said that that option doesn't really exist for genderIt doesn't. Gender transitioning is way harder and slower and more serious and also less accepted than hair dye.

gollark: A non mandatory one would be biased towards people who really care about whatever aspects of their identity it records.

gollark: I'd assume 10%ish, but nearby countries should be able to provide okay figures.

gollark: Okay, compare France or Germany or random nearby EU countries.

References

Moggi, Eugenio (July 1991). "Notions of computation and monads" (PDF). Information and Computation. 93 (1): 55–92. doi:10.1016/0890-5401(91)90052-4.
(ed.), Anca Muscholl (2014). Foundations of software science and computation structures : 17th (Aufl. 2014 ed.). [S.l.]: Springer. pp. 426–440. ISBN 978-3-642-54829-1.CS1 maint: extra text: authors list (link)

Anders Kock (1972). "Strong functors and monoidal monads" (PDF). Archiv der Mathematik. 23: 113–120. doi:10.1007/BF01304852.
Jean Goubault-Larrecq, Slawomir Lasota and David Nowak (2005). "Logical Relations for Monadic Types". Mathematical Structures in Computer Science. 18 (06): 1169. arXiv:cs/0511006. doi:10.1017/S0960129508007172.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Moggi, Eugenio (July 1991). "Notions of computation and monads" (PDF). Information and Computation. 93 (1): 55–92. doi:10.1016/0890-5401(91)90052-4.

[2] (ed.), Anca Muscholl (2014). Foundations of software science and computation structures : 17th (Aufl. 2014 ed.). [S.l.]: Springer. pp. 426–440. ISBN 978-3-642-54829-1.CS1 maint: extra text: authors list (link)