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.

↑ 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.

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.