# nLab Kleisli category

### Context

#### 2-Category theory

2-category theory

## Structures on 2-categories

#### Higher algebra

higher algebra

universal algebra

# Kleisli category

## Idea

Given a monad $T$ on some category $\mathcal{C}$, the Kleisli category of $T$ has as objects the objects of $\mathcal{C}$, but a morphism $X \to Y$ in the Kleisli category is a morphism $X \to T(Y)$ in $\mathcal{C}$. The monad structure induces a natural composition of such “$T$-shifted” morphisms.

Equivalently, the Kleisli category is the full subcategory of the Eilenberg–Moore category of $T$ on the free T-algebras (the free $T$modules).

## Definition

Let $\mathbf{T}=(T,\mu,\eta)$ be a monad in Cat, where $T:C\to C$ is an endofunctor with multiplication $\mu:T T\to T$ and unit $\eta:Id_C\to T$.

### 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 $\mathbf{T}$ the subcategory of the Eilenberg–Moore category $C^{\mathbf{T}}$ on the free $\mathbf{T}$-algebras.

###### 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 $C_{\mathbf{T}}$ has as objects the objects of $C$, and as morphisms $M\to N$ the elements of the hom-set $C(M,T(N))$, in other words morphisms of the form $M \to T(N)$ in $C$, called Kleisli morphisms.

Composition is given by the Kleisli composition rule $g\circ_{Kleisli} f = \mu_P\circ T(g)\circ f$ (as in the Grothendieck construction (here $M\stackrel{f}\to N\stackrel{g}\to P$).

###### Remark

More explicitly, this means that the Kleisli-composite of $f : x \to T y$ with $g : y \to T z$ is the morphism

$x \stackrel{f}{\to} T y \stackrel{T g}{\to} T T z \stackrel{\mu z}{\to} T z \,.$
###### Proof of equivalence

The equivalence between both presentations amounts to the functor $C_{T} \to C^{T}$ being full and faithful. This functor maps any object $X$ to $T(X)$, and any morphism $f \colon X \to T(Y)$ to $T(X) \stackrel{T(f)}{\to} T^2(Y) \stackrel{\mu_Y}{\to} T(Y)$.

Fullness holds because any morphism $g \colon T(X) \to T(Y)$ of algebras has as antecedent the composite $X \stackrel{\eta_X}{\to} T(X) \stackrel{g}{\to} T(Y)$. Indeed, the latter is mapped by the functor into $\mu_Y \circ T(g) \circ T(\eta_X)$, which because $g$ is a morphism of algebras is equal to $g \circ \mu_X \circ T(\eta_X)$, i.e., $g$.

Faithfulness holds as follows: if $\mu_Y \circ T(f) = \mu_Y \circ T(g)$, then precomposing by $\eta_X$ yields $\mu_Y \circ T(f) \circ \eta_X = \mu_Y \circ \eta_{T(Y)} \circ f = f$ and similarly for $g$, hence $f = g$.

###### 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 functional programming

In typed functional programming Kleisli composition is used to model functions with side-effects and computation. See at monad (in computer science) for more on this.

## References

• Jenö Szigeti, On limits and colimits in the Kleisli category, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 24 no. 4 (1983), p. 381-391 (NUMDAM)

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

• Marcelo Fiore and 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 in internal category theory is in

• Tomasz Brzeziński, Adrian Vazquez-Marquez, Internal Kleisli categories, Journal of Pure and Applied Algebra Volume 215, Issue 9, September 2011, Pages 2135–2147 (arXiv:0911.4048)

Discussion of Kleisli categories in type theory is in

• Alex Simpson, Recursive types in Kleisli Categories (pdf)

Revised on July 3, 2014 02:53:44 by Tom Hirschowitz (193.48.123.186)