In category theory, a strong monad over a

monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...

(''C'', ⊗, I) is a

monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', an ...

(''T'', η, μ) together with a

natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a na ...

''t''_''A,B'' : ''A'' ⊗ ''TB'' → ''T''(''A'' ⊗ ''B''), called (''tensorial'') ''strength'', such that the

diagrams A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-d ...

:, , :, and commute for every object ''A'', ''B'' and ''C'' (see Definition 3.2 in ). 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 In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict s ...

, a ''costrength'' natural transformation can be defined by :

t'_=T(\gamma_)\circ t_\circ\gamma_ : 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

. One interesting fact about commutative strong monads is that they are "the same as"

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

monoidal monads. More explicitly, * a commutative strong monad

(T,\eta,\mu,t)

defines a symmetric monoidal monad

(T,\eta,\mu,m)

by :

m_=\mu_\circ Tt'_\circ t_: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_=m_\circ(\eta_A\otimes 1_):A\otimes TB\to T(A\otimes B)

and the conversion between one and the other presentation is bijective.

References

* *{{cite journal , author = Jean Goubault-Larrecq, Slawomir Lasota and David Nowak , year = 2005 , title = Logical Relations for Monadic Types , doi = 10.1017/S0960129508007172 , journal = Mathematical Structures in Computer Science , volume = 18 , issue = 6 , pages = 1169 , arxiv = cs/0511006 , s2cid = 741758 Adjoint functors Monoidal categories