Strong monad explained

A strong monad is a mathematical object defined using category theory that is used in theoretical computer science. In technical terms, 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

$$t\text{'}\_=T(\backslash gamma\_)\backslash circ\; t\_\backslash circ\backslash gamma\_\; :\; TA\backslash otimes\; B\backslash to\; T(A\backslash otimes\; B).$$

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, η ,\mu,t) defines a symmetric monoidal monad (T, η ,\mu,m) by$$m\_=\backslash mu\_\backslash circ\; Tt\text{'}\_\backslash circ\; t\_:TA\backslash otimes\; TB\backslash to\; T(A\backslash otimes\; B)$$

• and conversely a symmetric monoidal monad

(T, η ,\mu,m) defines a commutative strong monad (T, η ,\mu,t) by$$t\_=m\_\backslash circ(\backslash eta\_A\backslash otimes\; 1\_):A\backslash otimes\; TB\backslash to\; T(A\backslash otimes\; B)$$and the conversion between one and the other presentation is bijective.

References

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

External links

• Strong monad at the nLab

Notes and References

1. Moggi. Eugenio. Notions of computation and monads. Information and Computation. July 1991. 93. 1. 55–92. 10.1016/0890-5401(91)90052-4. free.
2. Book: Anca. Muscholl. Anca Muscholl . Foundations of software science and computation structures : 17th. 2014. Springer. [S.l.]. 978-3-642-54829-1. 426–440. Aufl. 2014.