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 unit.svg,
Strong monad associative.svg,


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

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

Strong monad commutation.svg

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.


  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.