Kan extensions areuniversal constructs incategory theory, a branch ofmathematics. They are closely related toadjoints, but are also related tolimits andends. They are named afterDaniel M. Kan, who constructed certain (Kan) extensions usinglimits in 1960.

An early use of (what is now known as) a Kan extension from 1956 was inhomological algebra to computederived functors.

InCategories for the Working Mathematician,Saunders Mac Lane 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 toposets, it becomes a relatively familiar type of question onconstrained optimization.

A Kan extension proceeds from the data of three categories

${\displaystyle \mathbf{A},\mathbf{B},\mathbf{C}}$

and twofunctors

${\displaystyle X:\mathbf{A}\to \mathbf{C},F:\mathbf{A}\to \mathbf{B}}$,

and comes in two varieties: the "left" Kan extension and the "right" Kan extension of${\displaystyle X}$ along${\displaystyle F}$.

Abstractly, the functor${\displaystyle F}$ gives a pullback map${\displaystyle F^{\ast }:[B,C]\to [A,C]}$. When they exist, the left and right adjoints to${\displaystyle F^{\ast }}$ applied to${\displaystyle X}$ gives the left and right Kan extensions. Spelling the definition of adjoints out, we get the following definitions;

The right Kan extension amounts to finding the dashed arrow and thenatural transformation${\displaystyle ϵ}$ in the following diagram:

Formally, theright Kan extension of${\displaystyle X}$ along${\displaystyle F}$ consists of a functor${\displaystyle R:\mathbf{B}\to \mathbf{C}}$ and a natural transformation${\displaystyle ϵ:RF\to X}$ that isterminal with respect to this specification, in the sense that for any functor${\displaystyle M:\mathbf{B}\to \mathbf{C}}$ and natural transformation${\displaystyle \mu :MF\to X}$, a unique natural transformation${\displaystyle \delta :M\to R}$ is defined and fits into a commutative diagram:

where${\displaystyle {\delta }_F}$ is the natural transformation with${\displaystyle {\delta }_F(a)=\delta (Fa):MF(a)\to RF(a)}$ for anyobject${\displaystyle a}$ of${\displaystyle \mathbf{A}.}$

The functorR is often written${\displaystyle {\mathrm{Ran}}_{F}X}$.

As with the otheruniversal constructs incategory theory, the "left" version of the Kan extension isdual to the "right" one and is obtained by replacing all categories by theiropposites.

The effect of this on the description above is merely to reverse the direction of the natural transformations.

(Recall that anatural transformation${\displaystyle \tau }$ between the functors${\displaystyle F,G:\mathbf{C}\to \mathbf{D}}$ consists of having anarrow${\displaystyle \tau (a):F(a)\to G(a)}$ for every object${\displaystyle a}$ of${\displaystyle \mathbf{C}}$, satisfying a "naturality" property. When we pass to the opposite categories, the source and target of${\displaystyle \tau (a)}$ are swapped, causing${\displaystyle \tau }$ to act in the opposite direction).

This gives rise to the alternate description: theleft Kan extension of${\displaystyle X}$ along${\displaystyle F}$ consists of a functor${\displaystyle L:\mathbf{B}\to \mathbf{C}}$ and a natural transformation${\displaystyle \eta :X\to LF}$ that isinitial with respect to this specification, in the sense that for any other functor${\displaystyle M:\mathbf{B}\to \mathbf{C}}$ and natural transformation${\displaystyle \alpha :X\to MF}$, a unique natural transformation${\displaystyle \sigma :L\to M}$ exists and fits into a commutative diagram:

where${\displaystyle {\sigma }_F}$ is the natural transformation with${\displaystyle {\sigma }_F(a)=\sigma (Fa):LF(a)\to MF(a)}$ for any object${\displaystyle a}$ of${\displaystyle \mathbf{A}}$.

The functorL is often written${\displaystyle {\mathrm{Lan}}_{F}X}$.

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 uniqueup to uniqueisomorphism. In this case, that means that (for left Kan extensions) if${\displaystyle L,M}$ are two left Kan extensions of${\displaystyle X}$ along${\displaystyle F}$, and${\displaystyle \eta ,\alpha }$ are the corresponding transformations, then there exists a uniqueisomorphism of functors${\displaystyle \sigma :L\to M}$ such that the second diagram above commutes. Likewise for right Kan extensions.

Suppose${\displaystyle X:\mathbf{A}\to \mathbf{C}}$ and${\displaystyle F:\mathbf{A}\to \mathbf{B}}$ are two functors. IfA issmall andC is cocomplete, then there exists a left Kan extension${\displaystyle {\mathrm{Lan}}_{F}X}$ of${\displaystyle X}$ along${\displaystyle F}$, defined at each objectb ofB by

${\displaystyle ({\mathrm{Lan}}_{F}X)(b)=\underset{f:Fa\to b}{\underrightarrow{\lim }}X(a)}$

where the colimit is taken over thecomma category${\displaystyle (F\downarrow {\mathrm{const}}_b)}$, where${\displaystyle {\mathrm{const}}_b:\ast \to \mathbf{B},\ast \mapsto b}$ is the constant functor. Dually, ifA is small andC is complete, then right Kan extensions along${\displaystyle F}$ exist, and can be computed as the limit

${\displaystyle ({\mathrm{Ran}}_{F}X)(b)=\underset{Fa\leftarrow b}{\underleftarrow{\lim }}X(a)}$

over the comma category${\displaystyle ({\mathrm{const}}_b\downarrow F)}$.

Suppose${\displaystyle X:\mathbf{A}\to \mathbf{C}}$ and${\displaystyle F:\mathbf{A}\to \mathbf{B}}$ are two functors such that for all objectsa anda′ ofA and all objectsb ofB, thecopowers${\displaystyle \mathbf{B}(F{a}^{\prime },b)\cdot Xa}$ exist inC. Then the functorX has a left Kan extension${\displaystyle {\mathrm{Lan}}_{F}X}$ alongF, which is such that, for every objectb ofB,

${\displaystyle ({\mathrm{Lan}}_{F}X)b={\int }^a\mathbf{B}(Fa,b)\cdot Xa}$

when the abovecoend exists for every objectb ofB.

Dually, right Kan extensions can be computed by theend formula

${\displaystyle ({\mathrm{Ran}}_{F}X)b={\int }_{a}X{a}^{\mathbf{B}(b,Fa)}.}$

Thelimit of a functor${\displaystyle F:\mathbf{C}\to \mathbf{D}}$ can be expressed as a Kan extension by

${\displaystyle limF={\mathrm{Ran}}_{E}F}$

where${\displaystyle E}$ is the unique functor from${\displaystyle \mathbf{C}}$ to${\displaystyle \mathbf{1}}$ (thecategory with one object and one arrow, aterminal object in${\displaystyle \mathbf{C}\mathbf{a}\mathbf{t}}$). The colimit of${\displaystyle F}$ can be expressed similarly by

${\displaystyle \mathrm{colim}F={\mathrm{Lan}}_{E}F.}$

A functor${\displaystyle F:\mathbf{C}\to \mathbf{D}}$ possesses aleft adjointif and only if the right Kan extension of${\displaystyle \mathrm{Id}:\mathbf{C}\to \mathbf{C}}$ along${\displaystyle F}$ exists and is preserved by${\displaystyle F}$. In this case, a left adjoint is given by${\displaystyle {\mathrm{Ran}}_F\mathrm{Id}}$ and this Kan extension is even preserved by any functor${\displaystyle \mathbf{C}\to \mathbf{E}}$ whatsoever, i.e. is anabsolute Kan extension.

Dually, a right adjoint exists if and only if the left Kan extension of the identity along${\displaystyle F}$ exists and is preserved by${\displaystyle F}$.

Thecodensity monad of a functor${\displaystyle G:\mathbf{D}\to \mathbf{C}}$ is a right Kan extension ofG along itself.

• Cartan, Henri;Eilenberg, Samuel (1956).Homological algebra. Princeton Mathematical Series. Vol. 19. Princeton, New Jersey:Princeton University Press.Zbl 0075.24305.
• Mac Lane, Saunders (1998).Categories for the Working Mathematician.Graduate Texts in Mathematics. Vol. 5 (2nd ed.). New York, NY:Springer-Verlag.ISBN 0-387-98403-8.Zbl 0906.18001.
• Kelly, Gregory Maxwell (1982), "Chapter 4 Kan extensions",Basic concepts of enriched category theory(PDF), London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York,ISBN 0-521-28702-2,MR 0651714

• Model independent proof of colimit formula for left Kan extensions
• Kan extension at thenLab
• Kan extension as a limit: an example

• Adjoint functors
• Category theory

• Articles with short description
• Short description matches Wikidata