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symmetric monoidal (∞,1)-category of spectra
Given a monad on some category , then its Kleisli category is the full subcategory of the Eilenberg-Moore category of , hence the category of T-algebras, on those that are free T-algebras (free -modules).
Explicitly one may describe (Prop. below) the Kleisli category of (Def. ) to have as objects the objects of , and a morphism in the Kleisli category is a morphism in of the form in . The monad structure induces a natural composition of such “-shifted” morphisms.
The Kleisli category is also characterized by the following universal property:
Since every adjunction gives rise to a monad on the domain of its left adjoint, we might ask if every monad may be construed as arising from an adjunction. This is in fact true, and the initial such adjunction in the category of adjunctions for the given monad has the Kleisli category as the codomain of its left adjoint.
Let be a monad in Cat, where is an endofunctor with
multiplication ,
unit.
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 .
The Kleisli category of the monad is the full subcategory of the Eilenberg-Moore category on the free -algebras (Def. ).
If is the forgetful functor and is the free algebra functor , then the Kleisli category is simply the full subcategory of containing those objects in the image of .
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 of a monad on a category has:
as objects the objects of ,
as morphisms the morphisms of the form
in , called Kleisli morphisms;
and
composition of with is given by the Kleisli composition rule
the identity morphisms on is the Kleisli morphism which is the -unit .
(Kleisli equivalence)
The construction which sends a Kleisli morphism (1) to
constitutes a fully faithful functor
from the -Kleisli category (Def. ) to the category of -algebras, hence constitutes an equivalence of categories onto its essential image (the free -algebras).
(eg. Borceux (1994), Prop. 4.1.6)
To see that the functor is full, hence that is surjective, oberve that any homomorphism of algebras is the image of , as shown by the following commuting diagram:
Here the triangle on the left is the unit law of the monad, while the commutativity of the square is the fact that is a homomorphism of algebras.
To see that the functor is faithful, hence that is injective, notice that
by naturality of the unit combined with its unit law:
whence
This Kleisli composition plays an important role in computer science; for this, see the article at monad (in computer science).
If in addition to the given monad there is a comonad on the same category , equipped with a distributivity law (see there)
then there is a two-sided (“double”) Kleisli category whose objects are those of , and whose morphisms are morphisms in of the form
with two-sided Kelisli composition
given by the (co-)bind-operation on the factors connected by the distributivity transformation:
In the situation of Prop. , given in addition:
another monad on
then the usual compatibility of the one-sided-Kleisli category under monad transformations (see here) passes to the two-sided Kleisli category, in that
Consider the following diagram:
Here all squares commute by assumption on the monad transformation and hence the entire diagram commutes. Now the total top and right composite is the right hand side of (3), while the total left and bottom composite is the left hand side of (3), thus proving their equality.
Each Kleisli category comes with an adjunction
In terms of Kleisli morphisms:
In terms of free algebras:
In more general 2-categories the universal properties of Kleisli objects are dual to the universal properties of Eilenberg-Moore objects.
In particular, is initial in the category of adjunctions for (whereas is terminal). For a proof, see Category Theory in Context Proposition 5.2.12.
By the fact that the functor is left adjoint, it preserves all the colimits which exist in .
While , in general, doesn’t preserve all limits, one can say the following:
A limit in is preserved by if and only if it is preserved by the monad .
Let be a diagram in , and suppose its limit exists.
First, suppose that the limit of is preserved by . Then since is right adjoint, it preserves limits, and so the limit of is also preserved by .
Conversely, denote by the forgetful functor from the Eilenberg-Moore category of , and denote by the comparison functor. We have that factors as , and so . Now suppose that the limit of is preserved by . Since is monadic and monadic functors create limits, the limit is also preserved by the functor . Now since is a full inclusion, and full inclusions reflect limits, the limit is also preserved by the functor alone.
In typed functional programming, the Kleisli category is used to model call-by-value? functions with side effects and computation. Dually, the co-Kleisli category of a comonad may be used to model call-by-name? programming , see there.
Generally, see at monad (in computer science) for more on this.
(matrix multiplication as (co-)Kleisli composition)
See here.
The original articles:
Heinrich Kleisli, Every standard construction is induced by a pair of adjoint functors, Proc. Amer. Math. Soc. 16, AMS (1965) 544-546 [ISSN0002-9939, jstor:2034693]
Jean-Marie Maranda, On Fundamental Constructions and Adjoint Functors, Canadian Mathematical Bulletin 9 5 (1966) 581-591 [doi:10.4153/CMB-1966-072-9]
Early accounts (together with the Eilenberg-Moore category):
Fred Linton, §7 in: An outline of functorial semantics, in Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics 80, Springer (1969) 7-52 [doi:10.1007/BFb0083080]
Saunders MacLane, §VI.5 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971) [doi:10.1007/978-1-4757-4721-8]
Michael Barr, Charles Wells, §3.2 of: Toposes, Triples, and Theories, Grundlehren der math. Wissenschaften 278, Springer (1983) [TAC:12]
Jenö Szigeti, On limits and colimits in the Kleisli category, Cahiers de Topologie et Géométrie Différentielle Catégoriques 24 4 (1983) 381-391 [numdam:CTGDC_1983__24_4_381_0]
The equivalence of categories between the Kleisli category over a given monad with the co-Kleisli category of an adjoint comonad (if it exists):
The terminology “Kleisli triple” for a monad presented as an “extension system” and relation to computation with effects (see at monads in computer science):
Textbook account making explicit the Kleisli equivalence:
Lecture notes:
Discussion of cases where the inclusion of the Kleisli category into the Eilenberg-Moore category is a reflective subcategory:
Discussion of combined “double” or “two-sided” Kleisli categories, combining the Kleisli category of a monad with the co-Kleisli category of a comonad that distributes over it:
and generalization to 2-monads:
Discussion in internal category theory:
Discussion in a context of categorical systems theory:
Discussion of Kleisli categories in type theory is in
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