Kan extensions are
universal constructs in
category theory, a branch of
mathematics. They are closely related to
adjoints, but are also related to
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
and
ends
End, END, Ending, or variation, may refer to:
End
*In mathematics:
**End (category theory)
** End (topology)
**End (graph theory)
** End (group theory) (a subcase of the previous)
**End (endomorphism)
*In sports and games
** End (gridiron footbal ...
. They are named after
Daniel M. Kan, who constructed certain (Kan) extensions using
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
in 1960.
An early use of (what is now known as) a Kan extension from 1956 was in
homological algebra to compute
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
s.
In ''
Categories for the Working Mathematician
''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on ...
''
Saunders Mac Lane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.
Early life and education
Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...
titled a section "All Concepts Are Kan Extensions", and went on to write that
:The notion of Kan extensions subsumes all the other fundamental concepts of category theory.
Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
s, it becomes a relatively familiar type of question on
constrained optimization.
Definition
A Kan extension proceeds from the data of three categories
:
and two
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s
:
,
and comes in two varieties: the "left" Kan extension and the "right" Kan extension of
along
.
The right Kan extension amounts to finding the dashed arrow and the
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
in the following diagram:
:
Formally, the right Kan extension of
along
consists of a functor
and a natural transformation
that is couniversal with respect to the specification, in the sense that for any functor
and natural transformation
, a unique natural transformation
is defined and fits into a commutative diagram:
:
where
is the natural transformation with
for any
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
of
The functor ''R'' is often written
.
As with the other
universal constructs in
category theory, the "left" version of the Kan extension is
dual to the "right" one and is obtained by replacing all categories by their
opposites.
The effect of this on the description above is merely to reverse the direction of the natural transformations.
:(Recall that a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
between the functors
consists of having an
arrow for every object
of
, satisfying a "naturality" property. When we pass to the opposite categories, the source and target of
are swapped, causing
to act in the opposite direction).
This gives rise to the alternate description: the left Kan extension of
along
consists of a functor
and a natural transformation
that are universal with respect to this specification, in the sense that for any other functor
and natural transformation
, a unique natural transformation
exists and fits into a commutative diagram:
:
where
is the natural transformation with
for any object
of
.
The functor ''L'' is often written
.
The use of the word "the" (as in "the left Kan extension") is justified by the fact that, as with all universal constructions, if the object defined exists, then it is unique
up to unique
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. In this case, that means that (for left Kan extensions) if
are two left Kan extensions of
along
, and
are the corresponding transformations, then there exists a unique ''isomorphism'' of functors
such that the second diagram above commutes. Likewise for right Kan extensions.
Properties
Kan extensions as (co)limits
Suppose
and
are two functors. If A is
small and C is cocomplete, then there exists a left Kan extension
of
along
, defined at each object ''b'' of B by
:
where the colimit is taken over the
comma category , where
is the constant functor. Dually, if A is small and C is complete, then right Kan extensions along
exist, and can be computed as the limit
:
over the comma category
.
Kan extensions as (co)ends
Suppose
and
are two functors such that for all objects ''m'' and ''m'' of M and all objects ''c'' of C, the
copowers exist in A. Then the functor ''T'' has a left Kan extension ''L'' along ''K'', which is such that, for every object ''c'' of C,
:
when the above
coend exists for every object ''c'' of C.
Dually, right Kan extensions can be computed by the
end
End, END, Ending, or variation, may refer to:
End
*In mathematics:
** End (category theory)
** End (topology)
**End (graph theory)
** End (group theory) (a subcase of the previous)
**End (endomorphism)
*In sports and games
**End (gridiron footbal ...
formula
:
Limits as Kan extensions
The
limit of a functor
can be expressed as a Kan extension by
:
where
is the unique functor from
to
(the
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) ...
with one object and one arrow, a
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
in
). The colimit of
can be expressed similarly by
:
Adjoints as Kan extensions
A functor
possesses a
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
the right Kan extension of
along
exists and is preserved by
. In this case, a left adjoint is given by
and this Kan extension is even preserved by any functor
whatsoever, i.e. is an ''absolute'' Kan extension.
Dually, a right adjoint exists if and only if the left Kan extension of the identity along
exists and is preserved by
.
Applications
The
codensity monad of a functor
is a right Kan extension of ''G'' along itself.
References
*
*
External links
Model independent proof of colimit formula for left Kan extensions*
Kan extension as a limit: an example{{Category theory
Adjoint functors
Category theory