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Kleisli category

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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.

Formal definition[edit]

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.[1] We use very slightly different notation for this presentation. Given the same monad and category 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 define

Then the identity morphism in is

Extension operators and Kleisli triples[edit]

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 : XTY let

Composition in the Kleisli category CT can then be written

The extension operator satisfies the identities:

where f : XTY and g : YTZ. 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 tripleT, η, (–)#⟩, i.e.

  • A function ;
  • For each object in , a morphism ;
  • For each morphism in , a morphism

such that the above three equations for extension operators are satisfied.

Kleisli adjunction[edit]

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 FC → CT by

and a functor G : CTC 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⟩.

Showing that GF = T[edit]

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 . Q.E.D.


  1. ^ Mac Lane (1998). Categories for the Working Mathematician. p. 147.

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