Monad (category theory)

In category theory, a monad is a type of functor important in the theory of pairs of adjoint functors. If F and G are an adjoint pair of functors, with F left adjoint to G, then the composition FoG 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 GoF, is discussed under comonad.

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 T = FoG takes a group X and returns the free group Free(X) on the set underlying X. What we are given here consists of two observations: X ->T(X) by including any group X in Free(X) in the natural way. Further, T(T(X)) -> T(X) can be made out of a natural concatenation of 'strings of strings'. This amounts to two natural transformations I -> T, and ToT -> T. They will satisfy some axioms about identity and associativity based on the monoid axioms.

Hence in fact the name monad. Those axioms are taken as the definition of a general monad (not assumed a priori to be connected to an adjunction) on a category. Two constructions, the Kleisli category and the category of Eilenberg-Mac Lane algebras, are extremal solutions of the problem of constructing an adjunction starting with a given monad.

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.