Kleisli category

 ( Adjoint Functors ) 

In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of Free object 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.

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Formal definition Let ⟨ T, η, μ⟩ be a monad over a category C. The Kleisli category of C is the category C _T whose objects and morphisms are given by

\mathrm{Hom}_{\mathcal{C}_T}(X,Y) &= \mathrm{Hom}_{\mathcal{C}}(X,TY).\end{align} That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in C _T (but with codomain Y). Composition of morphisms in C _T is given by

$g\backslash circ\_T\; f\; =\; \backslash mu\_Z\; \backslash circ\; Tg\; \backslash circ\; f\; :\; X\; \backslash to\; T\; Y\; \backslash to\; T^2\; Z\; \backslash to\; T\; Z$

where f: X → T Y and g: Y → T Z. The identity morphism is given by the monad unit η:

$\backslash mathrm\{id\}\_X\; =\; \backslash eta\_X$.

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 $C$ as above, we associate with each object $X$ in $C$ a new object $X\_T$, and for each morphism $f\backslash colon\; X\backslash to\; TY$ in $C$ a morphism $f^*\backslash colon\; X\_T\backslash to\; Y\_T$. Together, these objects and morphisms form our category $C\_T$, where we define composition, also called Kleisli composition, by

$g^*\backslash circ\_T\; f^*\; =\; (\backslash mu\_Z\; \backslash circ\; Tg\; \backslash circ\; f)^*.$

Then the identity morphism in $C\_T$, the Kleisli identity, is

$\backslash mathrm\{id\}\_\{X\_T\}\; =\; (\backslash eta\_X)^*.$

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Extension 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 let

$f^\backslash sharp\; =\; \backslash mu\_Y\backslash circ\; Tf.$

Composition in the Kleisli category C _T can then be written

$g\backslash circ\_T\; f\; =\; g^\backslash sharp\; \backslash circ\; f.$

The extension operator satisfies the identities:

f^\sharp\circ\eta_X &= f\\ (g^\sharp\circ f)^\sharp &= g^\sharp \circ f^\sharp\end{align} where f : X → TY and g : Y → TZ. It follows trivially from these properties that Kleisli composition is associative and that η _X is the identity.

In fact, to give a monad is to give a Kleisli triple ⟨ T, η, (–)^#⟩, i.e.

• A function $T\backslash colon\; \backslash mathrm\{ob\}(C)\backslash to\; \backslash mathrm\{ob\}(C)$;
• For each object $A$ in $C$, a morphism $\backslash eta\_A\backslash colon\; A\backslash to\; T(A)$;
• For each morphism $f\backslash colon\; A\backslash to\; T(B)$ in $C$, a morphism $f^\backslash sharp\backslash colon\; T(A)\backslash to\; T(B)$

such that the above three equations for extension operators are satisfied.

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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 C _T be the associated Kleisli category. Using Mac Lane's notation mentioned in the “Formal definition” section above, define a functor F:  C →  C _T by

$FX\; =\; X\_T\backslash ;$

$F(f\backslash colon\; X\; \backslash to\; Y)\; =\; (\backslash eta\_Y\; \backslash circ\; f)^*$

and a functor G : C _T → C by

$GY\_T\; =\; TY\backslash ;$

$G(f^*\backslash colon\; X\_T\; \backslash to\; Y\_T)\; =\; \backslash mu\_Y\; \backslash circ\; Tf\backslash ;$

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

$\backslash varepsilon\_\{Y\_T\}\; =\; (\backslash mathrm\{id\}\_\{TY\})^*\; :\; (TY)\_T\; \backslash to\; Y\_T.$

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⟩.

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Showing that GF = T For any object X in category C:

$\backslash begin\{align\}$

(G \circ F)(X) &= G(F(X)) \\ &= G(X_T) \\ &= TX. \end{align}

For any $f\; :\; X\; \backslash to\; Y$ in category C:

$\backslash begin\{align\}$

(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{align} Since $(G\; \backslash circ\; F)(X)\; =\; TX$ is true for any object X in C and $(G\; \backslash circ\; F)(f)\; =\; Tf$ is true for any morphism f in C, then $G\; \backslash circ\; F\; =\; T$. Q.E.D.

• Saunders Mac Lane (1998). 9780387984032, Springer. ISBN 9780387984032
• (2025). 9780521834148, Cambridge University Press. ISBN 9780521834148
• Emily Riehl (2025). 9780486809038, Dover Publications. . ISBN 9780486809038

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