Kleisli category

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 C_T whose objects and morphisms are given by

{\begin{aligned}\mathrm {Obj} ({{\mathcal {C}}_{T}})&=\mathrm {Obj} ({\mathcal {C}}),\\\mathrm {Hom} _{{\mathcal {C}}_{T}}(X,Y)&=\mathrm {Hom} _{\mathcal {C}}(X,TY).\end{aligned}}

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

g\circ _{T}f=\mu _{Z}\circ Tg\circ f:X\to TY\to T^{2}Z\to TZ

where f: X → T Y and g: Y → T Z. The identity morphism is given by the monad unit η:

\mathrm {id} _{X}=\eta _{X}

.

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 $C$ as above, we associate with each object $X$ in $C$ a new object $X_{T}$ , and for each morphism $f\colon X\to TY$ in $C$ a morphism $f^{*}\colon X_{T}\to Y_{T}$ . Together, these objects and morphisms form our category $C_{T}$ , where we define

g^{*}\circ _{T}f^{*}=(\mu _{Z}\circ Tg\circ f)^{*}.

Then the identity morphism in $C_{T}$ is

\mathrm {id} _{X_{T}}=(\eta _{X})^{*}.

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 : X → TY let

f^{\sharp }=\mu _{Y}\circ Tf.

Composition in the Kleisli category C_T can then be written

g\circ _{T}f=g^{\sharp }\circ f.

The extension operator satisfies the identities:

{\begin{aligned}\eta _{X}^{\sharp }&=\mathrm {id} _{TX}\\f^{\sharp }\circ \eta _{X}&=f\\(g^{\sharp }\circ f)^{\sharp }&=g^{\sharp }\circ f^{\sharp }\end{aligned}}

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 $T\colon \mathrm {ob} (C)\to \mathrm {ob} (C)$ ;
For each object $A$ in $C$ , a morphism $\eta _{A}\colon A\to T(A)$ ;
For each morphism $f\colon A\to T(B)$ in $C$ , a morphism $f^{\sharp }\colon T(A)\to T(B)$

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 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