In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation tA,B : ATBT(AB), 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

.

A strong monad T is said to be commutative when the diagram

commutes for all objects and .[2]

One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,

  • a commutative strong monad defines a symmetric monoidal monad by
  • and conversely a symmetric monoidal monad defines a commutative strong monad by

and the conversion between one and the other presentation is bijective.

References

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