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.

References

doi:10.1016/0890-5401(91)90052-4
.

ISBN 978-3-642-54829-1
.

Anders Kock (1972). "Strong functors and monoidal monads" (PDF). Archiv der Mathematik. 23: 113–120.
S2CID 13246783
.

Jean Goubault-Larrecq, Slawomir Lasota and David Nowak (2005). "Logical Relations for Monadic Types". Mathematical Structures in Computer Science. 18 (6): 1169.
S2CID 741758
.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Strong_monad&oldid=1218229534"

[1] :10.1016/0890-5401(91)90052-4
.

[2] ISBN 978-3-642-54829-1
.

[1]

[2]