Kleisli category

In , a Kleisli category is a naturally associated to any T. It is equivalent to the category of free . The Kleisli category is one of two extremal solutions to the question Does every monad arise from an ? The other extremal solution is the . Kleisli categories are named for the mathematician Heinrich Kleisli. Let 〈T, η, μ〉 be a 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 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 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 〈T, η, (–)#〉, 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 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.

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Related publications (4)

Related concepts (2)

Twisting structures and morphisms up to strong homotopy

Kathryn Hess Bellwald

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the "strong homotopy" morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.

2019

Exponential Kleisli Monoids as Eilenberg-Moore Algebras

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Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

Springer2015

Order-adjoint monads and injective objects

Gavin Jay Seal

Given a monad T on Set whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category. then the monad induced by the new situation is Kock-Zoberlein. Injective objects in the category of Kleisli monoids with respect to the class of initial morphisms then characterize the objects of the Eilenberg-Moore category of T, a fact that allows us to recuperate a number of known results, and present some new ones. (C) 2009 Elsevier B.V. All rights reserved.

Elsevier2010

Kleisli category

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. Category theory is used in almost all areas of mathematics. In particular, numerous constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories.