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'', an ...
''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'', an ...
over a category ''C''. The Kleisli category of ''C'' is the category ''C''
''T'' 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 ''C''
''T'' (but with codomain ''Y''). Composition of morphisms in ''C''
''T'' 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
:
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
:
Composition in the Kleisli category ''C''
''T'' 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 ''η''
''X'' 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 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
:
:
and a functor ''G'' : ''C''
''T'' → ''C'' by
:
:
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
:
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'':
:
For any
in category ''C'':
:
Since
is true for any object ''X'' in ''C'' and
is true for any morphism ''f'' in ''C'', then
.
Q.E.D.
Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...
References
*
*
*
External links
*
{{Category theory
Adjoint functors
Categories in category theory