Kleisli Triple
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In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a Kleisli category is a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
naturally associated to any
monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', a ...
''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 In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is ...
. Kleisli categories are named for the mathematician
Heinrich Kleisli Heinrich Kleisli (; October 19, 1930 – April 5, 2011) was a Swiss 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 Quer ...
.


Formal definition

Let ⟨''T'', ''η'', ''μ''⟩ be a
monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', a ...
over a category ''C''. The Kleisli category of ''C'' is the category ''C''''T'' whose objects and morphisms are given by :\begin\mathrm() &= \mathrm(), \\ \mathrm_(X,Y) &= \mathrm_(X,TY).\end 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\circ_T f = \mu_Z \circ Tg \circ f : X \to T Y \to T^2 Z \to T Z where ''f: X → T Y'' and ''g: Y → T Z''. The identity morphism is given by the monad unit ''η'': :\mathrm_X = \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\colon X\to TY in C a morphism f^*\colon X_T\to Y_T. Together, these objects and morphisms form our category C_T, where we define :g^*\circ_T f^* = (\mu_Z \circ Tg \circ f)^*. Then the identity morphism in C_T is :\mathrm_ = (\eta_X)^*.


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^\sharp = \mu_Y\circ Tf. Composition in the Kleisli category ''C''''T'' can then be written :g\circ_T f = g^\sharp \circ f. The extension operator satisfies the identities: :\begin\eta_X^\sharp &= \mathrm_\\ f^\sharp\circ\eta_X &= f\\ (g^\sharp\circ f)^\sharp &= g^\sharp \circ f^\sharp\end 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\colon \mathrm(C)\to \mathrm(C); * For each object A in C, a morphism \eta_A\colon A\to T(A); * For each morphism f\colon A\to T(B) in C, a morphism f^\sharp\colon T(A)\to T(B) 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 ''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\; :F(f\colon X \to Y) = (\eta_Y \circ f)^* and a functor ''G'' : ''C''''T'' → ''C'' by :GY_T = TY\; :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 :\varepsilon_ = (\mathrm_)^* : (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''⟩.


Showing that ''GF'' = ''T''

For any object ''X'' in category ''C'': : \begin (G \circ F)(X) &= G(F(X)) \\ &= G(X_T) \\ &= TX. \end For any f : X \to Y in category ''C'': : \begin (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_ \circ Tf \\ &= Tf. \end Since (G \circ F)(X) = TX is true for any object ''X'' in ''C'' and (G \circ F)(f) = Tf is true for any morphism ''f'' in ''C'', then G \circ F = T. Q.E.D.


References

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External links

* {{Category theory Adjoint functors Categories in category theory