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

, a strong monad 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 ...

on a

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

with an additional natural transformation, called the strength, which governs how the monad interacts with the monoidal product. Strong monads play an important role in theoretical

computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...

where they are used to model computation with

side effects In medicine, a side effect is an effect of the use of a medicinal drug or other treatment, usually adverse but sometimes beneficial, that is unintended. Herbal and traditional medicines also have side effects. A drug or procedure usually used ...

Definition

A (left) strong monad is a

(''T'', η, μ) over a

(''C'', ⊗, I) 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'') ''left 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''.

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 ''right strength'' 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

Properties

The

Kleisli category In category theory, a Kleisli category is a category naturally associated to any monad ''T''. It is equivalent to the category of free ''T''-algebras. The Kleisli category is one of two extremal solutions to the question: "''Does every monad aris ...

of a commutative monad is symmetric monoidal in a canonical way, see corollary 7 in Guitart and corollary 4.3 in Power & Robison. 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 Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

monoidal monads. More explicitly, * a commutative strong monad

(T,\eta,\mu,t)

defines a symmetric monoidal monad

(T,\eta,\mu,m)

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)

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

{{Reflist

External links

Strong monad
at the

nLab The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab ...

Adjoint functors Monoidal categories