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Given a monad on some category , the Kleisli category of has as objects the objects of , but a morphism in the Kleisli category is a morphism in . The monad structure induces a natural composition of such “-shifted” morphisms.
Equivalently, the Kleisli category is the full subcategory of the Eilenberg–Moore category of on the free T-algebras (the free -modules).
Let be a monad in Cat, where is an endofunctor with multiplication and unit .
In terms of free algebras
A free -algebra over a monad (or free -module) is a -algebra (module) of the form , where the action is the component of multiplication transformation .
In terms of Kleisli morphisms
As another way of looking at this, we can keep the same objects as in but redefine the morphisms. This was the original Kleisli construction:
The Kleisli category has as objects the objects of , and as morphisms the elements of the hom-set , in other words morphisms of the form in , called Kleisli morphisms.
Composition is given by the Kleisli composition rule (as in the Grothendieck construction (here ).
Proof of equivalence
The equivalence between both presentations amounts to the functor being full and faithful. This functor maps any object to , and any morphism to .
Fullness holds because any morphism of algebras has as antecedent the composite . Indeed, the latter is mapped by the functor into , which because is a morphism of algebras is equal to , i.e., .
Faithfulness holds as follows: if , then precomposing by yields and similarly for , hence .
In more general 2-categories the universal properties of Kleisli objects are dual to the universal properties of Eilenberg–Moore objects?.
In functional programming
In typed functional programming Kleisli composition is used to model functions with side-effects and computation. See at monad (in computer science) for more on this.
- Jenö Szigeti, On limits and colimits in the Kleisli category, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 24 no. 4 (1983), p. 381-391 (NUMDAM)
Discussion of cases where the inclusion of the Kleisli category into the Eilenberg-Moore category is a reflective subcategory is in
- Marcelo Fiore, Matias Menni, Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads, Theory and Applications of Categories, Vol. 15, CT2004, No. 2, pp 40-65. (TAC)
Discussion in internal category theory is in
- Tomasz Brzeziński, Adrian Vazquez-Marquez, Internal Kleisli categories, Journal of Pure and Applied Algebra Volume 215, Issue 9, September 2011, Pages 2135–2147 (arXiv:0911.4048)
Discussion of Kleisli categories in type theory is in
- Alex Simpson, Recursive types in Kleisli Categories (pdf)