Strong monad

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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

Strong monad left unit.svg, Strong monad associative.svg,
Strong monad unit.svg, and Strong monad multiplication.svg

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[edit]

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

Strong monad commutation.svg

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.


  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.