nLab Kleisli category

Kleisli category

Context

2-Category theory

Higher algebra

Kleisli category

Idea

Given a monad TT on some category 𝒞\mathcal{C}, then its Kleisli category is the full subcategory of the Eilenberg-Moore category of TT, hence the category of T-algebras, on those that are free T-algebras (free TT-modules).

Explicitly one may describe (Prop. below) the Kleisli category of TT (Def. ) to have as objects the objects of 𝒞\mathcal{C}, and a morphism XYX \to Y in the Kleisli category is a morphism in 𝒞\mathcal{C} of the form XT(Y)X \to T(Y) in 𝒞\mathcal{C}. The monad structure induces a natural composition of such “TT-shifted” morphisms.

The Kleisli category is also characterized by the following universal property:

Since every adjunction gives rise to a monad on the domain of its left adjoint, we might ask if every monad may be construed as arising from an adjunction. This is in fact true, and the initial such adjunction in the category of adjunctions for the given monad has the Kleisli category as the codomain of its left adjoint.

Definition

Let T=(T,μ,η)\mathbf{T} = (T,\mu,\eta) be a monad in Cat, where T:𝒞𝒞T \colon \mathcal{C} \longrightarrow \mathcal{C} is an endofunctor with

  • multiplication μ:TTT\mu \colon T T \to T,

  • unitη:Id CT\eta \colon Id_C \to T.

In terms of free algebras

Definition

A free T\mathbf{T}-algebra over a monad (or free T\mathbf{T}-module) is a T\mathbf{T}-algebra (module) of the form (T(M),μ M)(T(M),\mu_M), where the action is the component of multiplication transformation μ M:T(T(M))T(M)\mu_M : T(T(M))\to T(M).

Definition

The Kleisli category C TC_{\mathbf{T}} of the monad T\mathbf{T} is the full subcategory of the Eilenberg-Moore category C TC^{\mathbf{T}} on the free T\mathbf{T}-algebras (Def. ).

Remark

If U:C TCU:C^{\mathbf{T}}\to C is the forgetful functor and F:CC TF: C\to C^{\mathbf{T}} is the free algebra functor F:M(TM,μ M)F: M\mapsto (T M,\mu_M), then the Kleisli category is simply the full subcategory of C TC^{\mathbf{T}} containing those objects in the image of FF.

In terms of Kleisli morphisms

As another way of looking at this, we can keep the same objects as in CC but redefine the morphisms. This was the original Kleisli construction:

Definition

The Kleisli category C TC_{\mathbf{T}} of a monad TT on a category 𝒞\mathcal{C} has:

  1. as objects the objects of 𝒞\mathcal{C},

  2. as morphisms MNM \to N the morphisms of the form

    (1)MT(N) M \longrightarrow{\;\;} T(N)

    in 𝒞\mathcal{C}, called Kleisli morphisms;

and

  • composition of MfTNM \xrightarrow{f} T N with NgTPN \xrightarrow{ g } T P is given by the Kleisli composition rule

    (2)g Kleislifμ PT(g)f:MfT(N)T(g)T(T(P))μ PT(P); g \circ_{Kleisli} f \;\coloneqq\; \mu_P \circ T(g) \circ f \;\colon\; M \overset{f}{\longrightarrow} T (N) \overset{T (g)}{\longrightarrow} T \big(T (P)\big) \overset{\mu_P}{\longrightarrow} T (P) \,;
  • the identity morphisms on MM is the Kleisli morphism which is the T T -unit Mη MTMM \xrightarrow{ \eta_M } T M.

Proposition

(Kleisli equivalence)
The construction which sends a Kleisli morphism XfTYX \xrightarrow{f} T Y (1) to

T(X)T(f)T 2(Y)μ YT(Y) T(X) \overset{T(f)}{\longrightarrow} T^2(Y) \overset{\mu_Y}{\longrightarrow} T(Y)

constitutes a full and faithful functor

𝒞 T𝒞 T \mathcal{C}_{T} \longrightarrow \mathcal{C}^{T}

from the TT-Kleisli category (Def. ) to the category of T T -algebras, hence constitutes an equivalence of categories onto its essential image which is that of free TT-algebras.

Proof

Fullness holds because any morphism g:T(X)T(Y)g \colon T(X) \to T(Y) of algebras has

as antecedent the composite Xη XT(X)gT(Y)X \stackrel{\eta_X}{\to} T(X) \stackrel{g}{\to} T(Y). Indeed, the latter is mapped by the functor into μ YT(g)T(η X)\mu_Y \circ T(g) \circ T(\eta_X), which is seen to be equal to gμ XT(η X)=gg \circ \mu_X \circ T(\eta_X) \;=\;g, using that gg is a homomorphism of algebras.

Faithfulness holds as follows: if μ YT(f)=μ YT(g)\mu_Y \circ T(f) = \mu_Y \circ T(g), then precomposing by η X\eta_X yields μ YT(f)η X=μ Yη T(Y)f=f\mu_Y \circ T(f) \circ \eta_X = \mu_Y \circ \eta_{T(Y)} \circ f = f and similarly for gg, hence f=gf = g.

Remark

This Kleisli composition plays an important role in computer science; for this, see the article at monad (in computer science).

Properties

Universal properties

In more general 2-categories the universal properties of Kleisli objects are dual to the universal properties of Eilenberg-Moore objects.

In particular, C TC_{\mathbf{T}} is initial in the category of adjunctions for T\mathbf{T} (whereas C TC^{\mathbf{T}} is terminal). For a proof, see Category Theory in Context Proposition 5.2.12.

Examples

General

Example

In typed functional programming, the Kleisli category is used to model call-by-value? functions with side effects and computation. Dually, the co-Kleisli category of a comonad may be used to model call-by-name? programming , see there.

Generally, see at monad (in computer science) for more on this.

Specific

Example

(matrix multiplication as (co-)Kleisli composition)
See here.

References

The original articles:

Early accounts (together with the Eilenberg-Moore category):

The equivalence of categories between the Kleisli category over a given monad with the co-Kleisli category of an adjoint comonad (if it exists):

The terminology “Kleisli triple” for a monad presented as an “extension system” and relation to computation with effects (see at monads in computer science):

Discussion of cases where the inclusion of the Kleisli category into the Eilenberg-Moore category is a reflective subcategory:

  • Marcelo Fiore, Matias Menni, Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads, Theory and Applications of Categories, Vol. 15, CT2004, No. 2, pp 40-65. (TAC)

Discussion of combined “double” or “two-sided” Kleisli categories, combining the Kleisli category of a monad with the co-Kleisli category of a comonad that distributes over it:

and generalization to 2-monads:

Discussion in internal category theory:

  • Tomasz Brzeziński, Adrian Vazquez-Marquez, Internal Kleisli categories, Journal of Pure and Applied Algebra 215 9 (2011) 213-147 [arXiv:0911.4048]

Discussion in a context of categorical systems theory:

Discussion of Kleisli categories in type theory is in

Last revised on November 22, 2022 at 12:26:24. See the history of this page for a list of all contributions to it.