The Kleisli category is one of two extremal solutions to the question: "Does every monad arise from an adjunction?"
The other extremal solution is the Eilenberg–Moore category.
Let ⟨T, η, μ⟩ be a monad over a category C. The Kleisli category of C is the category CT whose objects and morphisms are given by That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in CT (but with codomain Y).
The identity morphism is given by the monad unit η: An alternative way of writing this, which clarifies the category in which each object lives, is used by Mac Lane.
[1] We use very slightly different notation for this presentation.
Together, these objects and morphisms form our category
, the Kleisli identity, is Composition of Kleisli arrows can be expressed succinctly by means of the extension operator (–)# : Hom(X, TY) → Hom(TX, TY).
Given a monad ⟨T, η, μ⟩ over a category C and a morphism f : X → TY let Composition in the Kleisli category CT can then be written The extension operator satisfies the identities: where f : X → TY and g : Y → TZ.
It follows trivially from these properties that Kleisli composition is associative and that ηX is the identity.
In fact, to give a monad is to give a Kleisli triple ⟨T, η, (–)#⟩, i.e. such that the above three equations for extension operators are satisfied.
Kleisli categories were originally defined in order to show that every monad arises from an adjunction.
Using Mac Lane's notation mentioned in the “Formal definition” section above, define a functor F: C → CT by and a functor G : CT → C by One can show that F and G are indeed functors and that F is left adjoint to G. The counit of the adjunction is given by Finally, one can show that T = GF and μ = GεF so that ⟨T, η, μ⟩ is the monad associated to the adjunction ⟨F, G, η, ε⟩.