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| In [[category theory]], a '''Kleisli category''' is a [[category (mathematics)|category]] naturally associated to any [[monad (category theory)|monad]] ''T''. It is equivalent to the [[Eilenberg–Moore category|category of free ''T''-algebras]]. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise from an [[Adjunction (category theory)|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==
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| Let〈''T'', η, μ〉be a [[monad (category theory)|monad]] over a category ''C''. The '''Kleisli category''' of ''C'' is the category ''C''<sub>''T''</sub> whose objects and morphisms are given by
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| :<math>\begin{align}\mathrm{Obj}({\mathcal{C}_T}) &= \mathrm{Obj}({\mathcal{C}}), \\
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| \mathrm{Hom}_{\mathcal{C}_T}(X,Y) &= \mathrm{Hom}_{\mathcal{C}}(X,TY).\end{align}</math>
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| That is, every morphism ''f: X → T Y'' in ''C'' (with codomain ''TY'') can also be regarded as a morphism in ''C''<sub>''T''</sub> (but with codomain ''Y''). Composition of morphisms in ''C''<sub>''T''</sub> is given by
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| :<math>g\circ_T f = \mu_Z \circ Tg \circ f</math>
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| where ''f: X → T Y'' and ''g: Y → T Z''. The identity morphism is given by the monad unit η:
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| :<math>\mathrm{id}_X = \eta_X</math>.
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| An alternative way of writing this, which clarifies the category in which each object lives, is used by Mac Lane.<ref>Mac Lane(1998) p.147</ref> We use very slightly different notation for this presentation. Given the same monad and category <math>C</math> as above, we associate with each object <math>X</math> in <math>C</math> a new object <math>X_T</math>, and for each morphism <math>f:X\to TY</math> in <math>C</math> a morphism <math>f^*:X_T\to Y_T</math>. Together, these objects and morphisms form our category <math>C_T</math>, where we define
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| :<math>g^*\circ_T f^* = (\mu_Z \circ Tg \circ f)^*.</math>
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| Then the identity morphism in <math>C_T</math> is
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| :<math>\mathrm{id}_{X_T} = (\eta_X)^*.</math>
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| ==Extension operators and Kleisli triples==
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| 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
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| :<math>f^* = \mu_Y\circ Tf.</math>
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| Composition in the Kleisli category ''C''<sub>''T''</sub> can then be written
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| :<math>g\circ_T f = g^* \circ f.</math>
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| The extension operator satisfies the identities:
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| :<math>\begin{align}\eta_X^* &= \mathrm{id}_{TX}\\
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| f^*\circ\eta_X &= f\\
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| (g^*\circ f)^* &= g^* \circ f^*\end{align}</math>
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| where ''f'' : ''X'' → ''TY'' and ''g'' : ''Y'' → ''TZ''. It follows trivially from these properties that Kleisli composition is associative and that η<sub>''X''</sub> is the identity.
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| In fact, to give a monad is to give a ''Kleisli triple'', i.e.
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| * A function <math>T:\mathrm{ob}(C)\to \mathrm{ob}(C)</math>;
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| * For each object <math>A</math> in <math>C</math>, a morphism <math>\eta_A:A\to T(A)</math>;
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| * For each morphism <math>f:A\to T(B)</math> in <math>C</math>, a morphism <math>f^*:T(A)\to T(B)</math>
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| such that the above three equations for extension operators are satisfied.
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| ==Kleisli adjunction==
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| Kleisli categories were originally defined in order to show that every monad arises from an adjunction. That construction is as follows.
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| Let〈''T'', η, μ〉be a monad over a category ''C'' and let ''C''<sub>''T''</sub> be the associated Kleisli category. Define a functor ''F'' : ''C'' → ''C''<sub>''T''</sub> by
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| :<math>FX = X\;</math>
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| :<math>F(f : X \to Y) = \eta_Y \circ f</math>
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| and a functor ''G'' : ''C''<sub>''T''</sub> → ''C'' by
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| :<math>GY = TY\;</math>
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| :<math>G(f : X \to TY) = \mu_Y \circ Tf\;</math>
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| 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
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| :<math>\varepsilon_Y = \mathrm{id}_{TY}.</math>
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| Finally, one can show that ''T'' = ''GF'' and μ = ''G''ε''F'' so that 〈''T'', η, μ〉is the monad associated to the adjunction 〈''F'', ''G'', η, ε〉.
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| ==External links==
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| * {{nlab|id=Kleisli+category|title=Kleisli category}}
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| ==References==
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| {{reflist}}
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| * {{cite book | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=5 | location=New York, NY | publisher=[[Springer-Verlag]] | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 }}
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| * {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
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| [[Category:Adjoint functors]]
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Wilber Berryhill is the title his mothers and fathers gave him and he totally digs that title. To perform lacross is the factor I love most of all. Kentucky is where I've always been residing. For many years she's been working as a journey agent.
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