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[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

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.
^ 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.

Anders Kock (1972). "Strong functors and monoidal monads" (PDF). Archiv der Mathematik. 23: 113–120. doi:10.1007/BF01304852. S2CID 13246783.
Jean Goubault-Larrecq, Slawomir Lasota and David Nowak (2005). "Logical Relations for Monadic Types". Mathematical Structures in Computer Science. 18 (6): 1169. arXiv:cs/0511006. doi:10.1017/S0960129508007172. S2CID 741758.