Monad (category Theory) - Introduction

Introduction

If and are a pair of adjoint functors, with left adjoint to, then the composition is a monad. Therefore, a monad is an endofunctor. If and are inverse functors the corresponding monad is 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, is discussed under the dual theory of comonads.

The monad axioms can be seen at work in a simple example: let be the forgetful functor from the category Grp of groups to the category Set of sets. Then as we can take the free group functor.

This means that the monad

takes a set and returns the underlying set of the free group . In this situation, we are given two natural morphisms:

by including any set in in the natural way, as strings of length 1. Further,

can be made out of a natural concatenation or 'flattening' 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. 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 categories arising from partially ordered sets (with a single morphism from to iff ), then the formalism becomes much simpler: adjoint pairs are Galois connections and monads are closure operators.

Every monad arises from some adjunction, in fact typically from many adjunctions. Two constructions introduced below, the Kleisli category and the category of Eilenberg-Moore algebras, 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 a variety of algebras in universal algebra. 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-Moore algebras), so monads can also be seen as generalizing universal algebras. Even more generally, any adjunction is said to be monadic (or tripleable) if it shares this property of being (equivalent to) the Eilenberg-Moore category of its associated monad. Consequently Beck's monadicity theorem, which gives a criterion for monadicity, can be used to show that an arbitrary adjunction can be treated as a category of algebras in this way.

The notion of monad was invented by Godement in 1958 under the name standard construction. In the 1960s and 1970s, many people used the name triple. The term "monad", which is now standard, is due to Mac Lane.