Kleisli category
Encyclopedia
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
.
over a category C. The Kleisli category of C is the category CT 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 CT (but with codomain Y). Composition of morphisms in CT 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. 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
Composition in the Kleisli category CT 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 ηX is the identity.
In fact, to give a monad is to give a Kleisli triple, i.e.
such that the above three equations for extension operators are satisfied.
Let〈T, η, μ〉be a monad over a category C and let CT be the associated Kleisli category. Define a functor F : C → CT by
and a functor G : CT → 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〉.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a Kleisli category is a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
naturally associated to any monad
Monad (category theory)
In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...
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
Heinrich Kleisli
Heinrich Kleisli was a mathematician. He is the namesake of several constructions in category theory, including the Kleisli category and Kleisli triples. He is also the namesake of the Kleisli Query System, a tool for integration of heterogeneous databases developed at the University of...
.
Formal definition
Let〈T, η, μ〉be a monadMonad (category theory)
In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...
over a category C. The Kleisli category of C is the category CT 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 CT (but with codomain Y). Composition of morphisms in CT 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. 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 defineThen the identity morphism in
isExtension 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 letComposition in the Kleisli category CT 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 ηX is the identity.
In fact, to give a monad is to give a Kleisli triple, 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 CT be the associated Kleisli category. Define a functor F : C → CT by
and a functor G : CT → 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〉.