# nLab Kleisli category

Kleisli category

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

#### Higher algebra

higher algebra

universal algebra

# Kleisli category

## Idea

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

Explicitly one may describe (Prop. below) the Kleisli category of $T$ (Def. ) to have as objects the objects of $\mathcal{C}$, and a morphism $X \to Y$ in the Kleisli category is a morphism in $\mathcal{C}$ of the form $X \to T(Y)$ in $\mathcal{C}$. The monad structure induces a natural composition of such “$T$-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 $\mathbf{T} = (T,\mu,\eta)$ be a monad in Cat, where $T \colon \mathcal{C} \longrightarrow \mathcal{C}$ is an endofunctor with

• multiplication $\mu \colon T T \to T$,

• unit$\eta \colon Id_C \to T$.

### In terms of free algebras

###### Definition

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

###### Definition

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

###### Remark

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

### In terms of Kleisli morphisms

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

###### Definition

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

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

2. as morphisms $M \to N$ the morphisms of the form

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

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

and

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

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

###### Proposition

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

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

constitutes a fully faithful functor

$\mathcal{C}_{T} \xhookrightarrow{\phantom{--}} \mathcal{C}^{T}$

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

###### Proof

To see that the functor is full, hence that $f \mapsto \mu_Y \circ T(f)$ is surjective, oberve that any homomorphism $g \colon T(X) \to T(Y)$ of algebras is the image of $X \stackrel{\eta_X}{\to} T(X) \stackrel{g}{\to} T(Y)$, as shown by the following commuting diagram:

Here the triangle on the left is the unit law of the monad, while the commutativity of the square is the fact that $G$ is a homomorphism of algebras.

To see that the functor is faithful, hence that $f \mapsto \mu_Y \circ T(f)$ is injective, notice that

$\big( \mu_Y \circ T(f) \big) \circ \eta_X \;=\; f \,,$

by naturality of the unit $\eta_X$ combined with its unit law:

whence

$\mu_Y \circ T(f) \,=\, \mu_Y \circ T(g) \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \mu_Y \circ T(f) \circ \eta_X \,=\, \mu_Y \circ T(g) \circ \eta_X \;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\; f \,=\, g \,.$

###### Remark

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

### Two-sided Kleisli category

###### Proposition

If in addition to the given monad $\mathcal{E}$ there is a comonad $\mathcal{C}$ on the same category $\mathbf{C}$, equipped with a distributivity law (see there)

$distr^{\mathcal{C}, \mathcal{E}} \;\;\colon\;\; \mathcal{C} \big( \mathcal{E}(D) \big) \longrightarrow \mathcal{E} \big( \mathcal{C}(D) \big)$

then there is a two-sided (“double”) Kleisli category whose objects are those of $\mathbf{C}$, and whose morphisms $D_1 \to D_2$ are morphisms in $\mathbf{C}$ of the form

$prog_{12} \;\colon\; \mathcal{C}(D_1) \longrightarrow \mathcal{E}(D_2)$

with two-sided Kelisli composition

$prog_{12} \text{>=>} prog_{23} \;\; \colon \;\; \mathcal{C}(D_1) \longrightarrow \mathcal{E}(D_3)$

given by the (co-)bind-operation on the factors connected by the distributivity transformation:

(Brookes & Van Stone 1993 Thm. 2)

###### Proposition

In the situation of Prop. , given in addition:

1. $\mathcal{E}'$ another monad on $\mathbf{C}$

• also equipped with distributivity $distr^{\mathcal{C}, \mathcal{E}'} \,\colon\, \mathcal{C}\circ \mathcal{E}' \to \mathcal{E}' \circ \mathcal{C}$ over the given comonad $\mathcal{C}$,
2. a monad transformation $trans^{\mathcal{E} \to \mathcal{E}'} \,\colon\, \mathcal{E} \to \mathcal{E}'$

• which is compatible with the distributive laws in that that

then the usual compatibility of the one-sided-Kleisli category under monad transformations (see here) passes to the two-sided Kleisli category, in that

(3)$\big( trans^{ \mathcal{E} \to \mathcal{E}' }_{D_2} \circ prog_{12} \big) \;\; \text{>=>} \;\; \big( trans^{ \mathcal{E} \to \mathcal{E}' }_{D_3} \circ prog_{23} \big) \;\;\; = \;\;\; trans^{ \mathcal{E} \to \mathcal{E}' }_{D_3} \circ \big( prog_{12} \;\text{>=>}\; prog_{23} \big) \,.$

###### Proof

Consider the following diagram:

Here all squares commute by assumption on the monad transformation and hence the entire diagram commutes. Now the total top and right composite is the right hand side of (3), while the total left and bottom composite is the left hand side of (3), thus proving their equality.

## 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, $C_{\mathbf{T}}$ is initial in the category of adjunctions for $\mathbf{T}$ (whereas $C^{\mathbf{T}}$ 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):

Textbook account making explicit the Kleisli equivalence:

Lecture notes:

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: