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 arise from an adjunction? The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli.
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). Composition of morphisms in CT is given by
where f: X → T Y and g: Y → T Z. 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. as above, we associate with each object in a new object , and for each morphism in a morphism . Together, these objects and morphisms form our category , where we defineWe use very slightly different notation for this presentation. Given the same monad and category
Then the identity morphism in 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. That construction is as follows.
Let ⟨T, η, μ⟩ be a monad over a category C and let CT be the associated Kleisli category. 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⟩.
For any object X in category C:
For any in category C:
Since is true for any object X in C and is true for any morphism f in C, then .
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