Kleisli category

<h2 id="formal-definition">Formal definition</h2>
Let ⟨T, η, μ⟩ be a <a href="/facts/Monad_(category_theory)/6fX10Qrx">monad</a> over a category C. The Kleisli category of C is the category CT whose objects and morphisms are given by

O
                  b
                  j
                
                (
                
                  
                    
                      
                        C
                      
                    
                    
                      T
                    
                  
                
                )
              
              
                
                =
                
                  O
                  b
                  j
                
                (
                
                  
                    C
                  
                
                )
                ,
              
            
            
              
                
                  
                    H
                    o
                    m
                  
                  
                    
                      
                        
                          C
                        
                      
                      
                        T
                      
                    
                  
                
                (
                X
                ,
                Y
                )
              
              
                
                =
                
                  
                    H
                    o
                    m
                  
                  
                    
                      C
                    
                  
                
                (
                X
                ,
                T
                Y
                )
                .
              
            
          
        
      
    
    {\displaystyle {\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 CT (but with codomain Y). Composition of morphisms in CT is given by

g
        
          ∘
          
            T
          
        
        f
        =
        
          μ
          
            Z
          
        
        ∘
        T
        g
        ∘
        f
        :
        X
        →
        T
        Y
        →
        
          T
          
            2
          
        
        Z
        →
        T
        Z
      
    
    {\displaystyle 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:

i
 d
 
 
 X
 
 
 =
 
 η
 
 X
 
 
 
 
 {\displaystyle \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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> We use very slightly different notation for this presentation. Given the same monad and category 
 
 
 
 C
 
 
 {\displaystyle C}
 
 as above, we associate with each object 
 
 
 
 X
 
 
 {\displaystyle X}
 
 in 
 
 
 
 C
 
 
 {\displaystyle C}
 
 a new object 
 
 
 
 
 X
 
 T
 
 
 
 
 {\displaystyle X_{T}}
 
, and for each morphism 
 
 
 
 f
 :
 X
 →
 T
 Y
 
 
 {\displaystyle f\colon X\to TY}
 
 in 
 
 
 
 C
 
 
 {\displaystyle C}
 
 a morphism 
 
 
 
 
 f
 
 ∗
 
 
 :
 
 X
 
 T
 
 
 →
 
 Y
 
 T
 
 
 
 
 {\displaystyle f^{*}\colon X_{T}\to Y_{T}}
 
. Together, these objects and morphisms form our category 
 
 
 
 
 C
 
 T
 
 
 
 
 {\displaystyle C_{T}}
 
, where we define composition, also called Kleisli composition, by

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

Then the identity morphism in 
 
 
 
 
 C
 
 T
 
 
 
 
 {\displaystyle C_{T}}
 
, the Kleisli identity, is

i
            d
          
          
            
              X
              
                T
              
            
          
        
        =
        (
        
          η
          
            X
          
        
        
          )
          
            ∗
          
        
        .
      
    
    {\displaystyle \mathrm {id} _{X_{T}}=(\eta _{X})^{*}.}

<h2 id="extension-operators-and-kleisli-triples">Extension operators and Kleisli triples</h2>
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
          
            ♯
          
        
        =
        
          μ
          
            Y
          
        
        ∘
        T
        f
        .
      
    
    {\displaystyle f^{\sharp }=\mu _{Y}\circ Tf.}

Composition in the Kleisli category CT can then be written

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

The extension operator satisfies the identities:

η
                  
                    X
                  
                  
                    ♯
                  
                
              
              
                
                =
                
                  
                    i
                    d
                  
                  
                    T
                    X
                  
                
              
            
            
              
                
                  f
                  
                    ♯
                  
                
                ∘
                
                  η
                  
                    X
                  
                
              
              
                
                =
                f
              
            
            
              
                (
                
                  g
                  
                    ♯
                  
                
                ∘
                f
                
                  )
                  
                    ♯
                  
                
              
              
                
                =
                
                  g
                  
                    ♯
                  
                
                ∘
                
                  f
                  
                    ♯
                  
                
              
            
          
        
      
    
    {\displaystyle {\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 ηX is the identity.
In fact, to give a monad is to give a Kleisli triple ⟨T, η, (–)#⟩, i.e. 

<ul><li>A function 
 
 
 
 T
 :
 
 o
 b
 
 (
 C
 )
 →
 
 o
 b
 
 (
 C
 )
 
 
 {\displaystyle T\colon \mathrm {ob} (C)\to \mathrm {ob} (C)}
 
;</li>
<li>For each object 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in 
 
 
 
 C
 
 
 {\displaystyle C}
 
, a morphism 
 
 
 
 
 η
 
 A
 
 
 :
 A
 →
 T
 (
 A
 )
 
 
 {\displaystyle \eta _{A}\colon A\to T(A)}
 
;</li>
<li>For each morphism 
 
 
 
 f
 :
 A
 →
 T
 (
 B
 )
 
 
 {\displaystyle f\colon A\to T(B)}
 
 in 
 
 
 
 C
 
 
 {\displaystyle C}
 
, a morphism 
 
 
 
 
 f
 
 ♯
 
 
 :
 T
 (
 A
 )
 →
 T
 (
 B
 )
 
 
 {\displaystyle f^{\sharp }\colon T(A)\to T(B)}
 
</li></ul>
such that the above three equations for extension operators are satisfied.

<h2 id="kleisli-adjunction">Kleisli adjunction</h2>
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. Using Mac Lane's notation mentioned in the “Formal definition” section above, define a functor F: C → CT by

F
        X
        =
        
          X
          
            T
          
        
        
      
    
    {\displaystyle FX=X_{T}\;}

F
        (
        f
        :
        X
        →
        Y
        )
        =
        (
        
          η
          
            Y
          
        
        ∘
        f
        
          )
          
            ∗
          
        
      
    
    {\displaystyle F(f\colon X\to Y)=(\eta _{Y}\circ f)^{*}}

and a functor G : CT → C by

G
        
          Y
          
            T
          
        
        =
        T
        Y
        
      
    
    {\displaystyle GY_{T}=TY\;}

G
        (
        
          f
          
            ∗
          
        
        :
        
          X
          
            T
          
        
        →
        
          Y
          
            T
          
        
        )
        =
        
          μ
          
            Y
          
        
        ∘
        T
        f
        
      
    
    {\displaystyle G(f^{*}\colon X_{T}\to Y_{T})=\mu _{Y}\circ Tf\;}

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

ε
          
            
              Y
              
                T
              
            
          
        
        =
        (
        
          
            i
            d
          
          
            T
            Y
          
        
        
          )
          
            ∗
          
        
        :
        (
        T
        Y
        
          )
          
            T
          
        
        →
        
          Y
          
            T
          
        
        .
      
    
    {\displaystyle \varepsilon _{Y_{T}}=(\mathrm {id} _{TY})^{*}:(TY)_{T}\to Y_{T}.}

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⟩.

<h3>Showing that GF = T</h3>
For any object X in category C:

(
                G
                ∘
                F
                )
                (
                X
                )
              
              
                
                =
                G
                (
                F
                (
                X
                )
                )
              
            
            
              
              
                
                =
                G
                (
                
                  X
                  
                    T
                  
                
                )
              
            
            
              
              
                
                =
                T
                X
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(G\circ F)(X)&=G(F(X))\\&=G(X_{T})\\&=TX.\end{aligned}}}

For any 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 in category C:

(
                G
                ∘
                F
                )
                (
                f
                )
              
              
                
                =
                G
                (
                F
                (
                f
                )
                )
              
            
            
              
              
                
                =
                G
                (
                (
                
                  η
                  
                    Y
                  
                
                ∘
                f
                
                  )
                  
                    ∗
                  
                
                )
              
            
            
              
              
                
                =
                
                  μ
                  
                    Y
                  
                
                ∘
                T
                (
                
                  η
                  
                    Y
                  
                
                ∘
                f
                )
              
            
            
              
              
                
                =
                
                  μ
                  
                    Y
                  
                
                ∘
                T
                
                  η
                  
                    Y
                  
                
                ∘
                T
                f
              
            
            
              
              
                
                =
                
                  
                    id
                  
                  
                    T
                    Y
                  
                
                ∘
                T
                f
              
            
            
              
              
                
                =
                T
                f
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(G\circ F)(f)&=G(F(f))\\&=G((\eta _{Y}\circ f)^{*})\\&=\mu _{Y}\circ T(\eta _{Y}\circ f)\\&=\mu _{Y}\circ T\eta _{Y}\circ Tf\\&={\text{id}}_{TY}\circ Tf\\&=Tf.\end{aligned}}}

Since 
 
 
 
 (
 G
 ∘
 F
 )
 (
 X
 )
 =
 T
 X
 
 
 {\displaystyle (G\circ F)(X)=TX}
 
 is true for any object X in C and 
 
 
 
 (
 G
 ∘
 F
 )
 (
 f
 )
 =
 T
 f
 
 
 {\displaystyle (G\circ F)(f)=Tf}
 
 is true for any morphism f in C, then 
 
 
 
 G
 ∘
 F
 =
 T
 
 
 {\displaystyle G\circ F=T}
 
. <a href="/facts/Q.E.D./B8I8jPwT">Q.E.D.</a>

<ul><li><a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Mac Lane, Saunders</a> (1998). <a href="/facts/Categories_for_the_Working_Mathematician/SLQDat9B">Categories for the Working Mathematician</a>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 5 (2nd ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98403-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0906.18001">0906.18001</a>.</li>
<li>Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-83414-7. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1034.18001">1034.18001</a>.</li>
<li><a href="/facts/Emily_Riehl/sPtNtfrP">Riehl, Emily</a> (2016). <a href="https://math.jhu.edu/~eriehl/context.pdf#page=153">Category Theory in Context</a> (PDF). Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-80903-8. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1006743127">1006743127</a>.</li>
<li><a href="/facts/Jacques_Riguet/z8aRuzPk">Riguet, Jacques</a>; Guitart, Rene (1992). <a href="http://www.numdam.org/item/CTGDC_1992__33_3_261_0">"Enveloppe Karoubienne et categorie de Kleisli"</a>. Cahiers de Topologie et Géométrie Différentielle Catégoriques. 33 (3): 261–6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1186950">1186950</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0767.18008">0767.18008</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://ncatlab.org/nlab/show/Kleisli+category">Kleisli category</a> at the <a href="/facts/NLab/lpPvRmgo">nLab</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Mac Lane (1998). Categories for the Working Mathematician. p. 147. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Kleisli category open-in-new

Kleisli category