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If F and G are an adjoint pair of functors, with F left adjoint to G, then the composition GoF will be a monad. Note that therefore a monad is a functor from a category to itself; and that if F and G were actually inverses as functors the corresponding monad would be the identity functor. In general adjunctions are not equivalences — they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The other half of the theory, of what can be learned likewise from consideration of FoG, is discussed under the dual theory of comonads.
The monad axioms can be seen at work in a simple example: let G be the forgetful functor from the category Group of groups to the category Set of sets. Then as F we can take the free group functor.
This means that the monad
takes a set X and returns the underlying set of the free group Free(X). In this situation, we are given two natural morphisms:
by including any set X in Free(X) in the natural way, as strings of length 1. Further,
can be made out of a natural concatenation of 'strings of strings'. This amounts to two natural transformations
and
They will satisfy some axioms about identity and associativity that result from the adjunction properties.
Those axioms are formally similar to the monoid axioms. (In fact, a monad can be seen as a " monoid object" in the functor categoryCategory theory In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. Functor categories are of interest for two main re of all functors from a given category to itself.) They are taken as the definition of a general monad (not assumed a priori to be connected to an adjunction) on a category.
If we specialize to the category arising from a partially ordered setIn mathematics, partially ordered sets or posets for short, are special binary relations which formalize the intuitive concept of an ordering. Partially ordered sets are studied in order theory and a much more detailed introduction to the field can be fou (P, ≤) (with a single morphism from x to y iffIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P x ≤ y), then the formalism becomes much simpler: adjoint pairs are Galois connectionIn mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets ("posets"). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory.s and monads are closure operators.
Every monad arises from some adjunction, in fact typically from many adjunctions. Two constructions, the Kleisli category and the category of Eilenberg-Mac Lane algebra s, are extremal solutions of the problem of constructing an adjunction that gives rise to a given monad.
The example about free groups given above can be generalized to any type of algebra in the sense of universal algebraUniversal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Basic idea From the point of view of universal algebra, an algebra is a set A together with a collection of operations on A. An n- ary operation on A. Thus, every such type of algebra gives rise to a monad on the category of sets. Importantly, the algebra type can be recovered from the monad (as the category of Eilenberg-Mac Lane algebras), so monads can also be seen as generalizing universal algebra.
While monads are quite common, making them explicit is less so (the language belongs to the school of Mac Lane, and has rarely been used in the school of Grothendieck, which prefers to write out monads and comonads longhand). In categorical logic, an analogy has been drawn between the monad-comonad theory, and modal logic.