category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, 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 and r ...

(''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 natur ...

''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- ...

:, , :, and commute for every object ''A'', ''B'' and ''C'' (see Definition 3.2 in ). If the monoidal category (''C'', ⊗, I) is

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

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 sen ...

, 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 definiti ...

monoidal monad In category theory, a monoidal monad (T,\eta,\mu,T_,T_0) is a monad (T,\eta,\mu) on a monoidal category (C,\otimes,I) such that the functor T:(C,\otimes,I)\to(C,\otimes,I) is a lax monoidal functor and the natural transformations \eta and \mu are m ...

s. 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