Definitions
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symmetric monoidal (∞,1)-category of spectra
Given a monad $T$ on some category $\mathcal{C}$, then its Kleisli category is the full subcategory of the Eilenberg-Moore category of $T$, hence the category of T-algebras, on those that are free T-algebras (free $T$-modules).
Explicitly one may describe (Prop. below) the Kleisli category of $T$ (Def. ) to have as objects the objects of $\mathcal{C}$, and a morphism $X \to Y$ in the Kleisli category is a morphism in $\mathcal{C}$ of the form $X \to T(Y)$ in $\mathcal{C}$. The monad structure induces a natural composition of such “$T$-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 $\mathbf{T} = (T,\mu,\eta)$ be a monad in Cat, where $T \colon \mathcal{C} \longrightarrow \mathcal{C}$ is an endofunctor with
multiplication $\mu \colon T T \to T$,
unit$\eta \colon Id_C \to T$.
A free $\mathbf{T}$-algebra over a monad (or free $\mathbf{T}$-module) is a $\mathbf{T}$-algebra (module) of the form $(T(M),\mu_M)$, where the action is the component of multiplication transformation $\mu_M : T(T(M))\to T(M)$.
The Kleisli category $C_{\mathbf{T}}$ of the monad $\mathbf{T}$ is the full subcategory of the Eilenberg-Moore category $C^{\mathbf{T}}$ on the free $\mathbf{T}$-algebras (Def. ).
If $U:C^{\mathbf{T}}\to C$ is the forgetful functor and $F: C\to C^{\mathbf{T}}$ is the free algebra functor $F: M\mapsto (T M,\mu_M)$, then the Kleisli category is simply the full subcategory of $C^{\mathbf{T}}$ containing those objects in the image of $F$.
As another way of looking at this, we can keep the same objects as in $C$ but redefine the morphisms. This was the original Kleisli construction:
The Kleisli category $C_{\mathbf{T}}$ of a monad $T$ on a category $\mathcal{C}$ has:
as objects the objects of $\mathcal{C}$,
as morphisms $M \to N$ the morphisms of the form
in $\mathcal{C}$, called Kleisli morphisms;
and
composition of $M \xrightarrow{f} T N$ with $N \xrightarrow{ g } T P$ is given by the Kleisli composition rule
the identity morphisms on $M$ is the Kleisli morphism which is the $T$-unit $M \xrightarrow{ \eta_M } T M$.
(Kleisli equivalence)
The construction which sends a Kleisli morphism $X \xrightarrow{f} T Y$ (1) to
constitutes a full and faithful functor
from the $T$-Kleisli category (Def. ) to the category of $T$-algebras, hence constitutes an equivalence of categories onto its essential image which is that of free $T$-algebras.
Fullness holds because any morphism $g \colon T(X) \to T(Y)$ of algebras has
as antecedent the composite $X \stackrel{\eta_X}{\to} T(X) \stackrel{g}{\to} T(Y)$. Indeed, the latter is mapped by the functor into $\mu_Y \circ T(g) \circ T(\eta_X)$, which is seen to be equal to $g \circ \mu_X \circ T(\eta_X) \;=\;g$, using that $g$ is a homomorphism of algebras.
Faithfulness holds as follows: if $\mu_Y \circ T(f) = \mu_Y \circ T(g)$, then precomposing by $\eta_X$ yields $\mu_Y \circ T(f) \circ \eta_X = \mu_Y \circ \eta_{T(Y)} \circ f = f$ and similarly for $g$, hence $f = g$.
This Kleisli composition plays an important role in computer science; for this, see the article at monad (in computer science).
In more general 2-categories the universal properties of Kleisli objects are dual to the universal properties of Eilenberg-Moore objects.
In particular, $C_{\mathbf{T}}$ is initial in the category of adjunctions for $\mathbf{T}$ (whereas $C^{\mathbf{T}}$ is terminal). For a proof, see Category Theory in Context Proposition 5.2.12.
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]
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]
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]
The equivalence of categories between the Kleisli category over a given monad with the co-Kleisli category of an adjoint comonad (if it exists):
Volume 133 1 (1990) 79-82 [doi:10.1016/0021-8693(90)90069-Z]
The terminology “Kleisli triple” for a monad presented as an “extension system” and relation to computation with effects (see at monads in computer science):
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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