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

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 .

## References

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

- Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. 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.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001. - Jacques Riguet & Rene Guitart (1992) Enveloppe Karoubienne et categorie de Kleisli,
*Cahiers de Topologie et Géométrie Différentielle Catégoriques*33(3) : 261–6, via Numdam.org