In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free Talgebras. 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.
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Category Theory 3.2: Kleisli category

Kleisli categories and probability  02  The Kleisli category of a monad

Kleisli categories and probability  01  The Giry monad

The Kleisli Category for a Monad

Kleisli categories and probability  03  Markov kernels
Transcription
Formal definition
Let ⟨T, η, μ⟩ be a monad over a category C. The Kleisli category of C is the category C_{T} 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 C_{T} (but with codomain Y). Composition of morphisms in C_{T} 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
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 C_{T} 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 η_{X} is the identity.
In fact, to give a monad is to give a Kleisli triple ⟨T, 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
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 C_{T} be the associated Kleisli category. Using Mac Lane's notation mentioned in the “Formal definition” section above, define a functor F: C → C_{T} by
and a functor G : C_{T} → 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⟩.
Showing that GF = T
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.
References
 ^ Mac Lane (1998). Categories for the Working Mathematician. p. 147.
 Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer. ISBN 0387984038. Zbl 0906.18001.
 Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge University Press. ISBN 0521834147. Zbl 1034.18001.
 Riehl, Emily (2016). Category Theory in Context (PDF). Dover Publications. ISBN 9780486809038. OCLC 1006743127.
 Riguet, Jacques; Guitart, Rene (1992). "Enveloppe Karoubienne et categorie de Kleisli". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 33 (3): 261–6. MR 1186950. Zbl 0767.18008.
External links
 Kleisli category at the nLab