Strong monad

Last updated October 23, 2024

In category theory, a strong monad is a monad on a monoidal category with an additional natural transformation, called the strength, which governs how the monad interacts with the monoidal product.

Definition

A (left) strong monad is a monad (T, η, μ) over a monoidal category (C, ⊗, I) together with a natural transformation t_A,B : A ⊗ TB → T(A ⊗ B), called (tensorial) left strength, such that the diagrams

,

,

, and

commute for every object A, B and C.

Commutative strong monads

For every strong monad T on a symmetric monoidal category, a right strength 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 $B$ .

Properties

The Kleisli category of a commutative monad is symmetric monoidal in a canonical way, see corollary 7 in Guitart^[2] and corollary 4.3 in Power & Robison^[3]. When a monad is strong but not necessarily commutative, its Kleisli category is a premonoidal category.

One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads.^[4] 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.

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References

↑ 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 .
↑ Guitart, René (1980). "Tenseurs et machines". Cahiers de topologie et géométrie différentielle. 21 (1): 5–62. ISSN 2681-2398.
↑ Power, John; Robinson, Edmund (October 1997). "Premonoidal categories and notions of computation". Mathematical Structures in Computer Science. 7 (5): 453–468. doi:10.1017/S0960129597002375. ISSN 0960-1295.
↑ Kock, Anders (1972-12-01). "Strong functors and monoidal monads". Archiv der Mathematik. 23 (1): 113–120. doi:10.1007/BF01304852. ISSN 1420-8938.

External links

Strong monad at the nLab

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[Moggi91-1] 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] Guitart, René (1980). "Tenseurs et machines". Cahiers de topologie et géométrie différentielle. 21 (1): 5–62. ISSN 2681-2398.

[3] Power, John; Robinson, Edmund (October 1997). "Premonoidal categories and notions of computation". Mathematical Structures in Computer Science. 7 (5): 453–468. doi:10.1017/S0960129597002375. ISSN 0960-1295.

[4] Kock, Anders (1972-12-01). "Strong functors and monoidal monads". Archiv der Mathematik. 23 (1): 113–120. doi:10.1007/BF01304852. ISSN 1420-8938.

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