Kan extension

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Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named for Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.

An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors.

In Categories 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."

The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on 'constrained optimization'.

1 Definition
- 1.1 Left Kan extension
- 1.2 Right Kan extension
2 Properties
3 References

[edit] Definition

As with the other universal constructs in category theory, there are two kinds of Kan extensions, which are dual to one another.

The left Kan extension is so named because, in its definition, the required unique morphism for an arbitrary candidate functor has the left Kan extension as the domain functor, i.e. usually written on the left, e.g.

$\sigma:L \rightarrow M$ ,

where $L\$ is the left Kan extension, $M\$ is the candidate functor, and $\sigma\$ is a natural transformation between them.

Dually, the right Kan extension is so named because, in its definition, the required unique morphism for an arbitrary candidate functor has the right Kan extension as the codomain functor, i.e. usually written on the right, e.g.

$\delta:M \rightarrow R$ ,

where $M\$ is the candidate functor, $R\$ is the right Kan extension, and $\delta\$ is a natural transformation between them.

[edit] Left Kan extension

In this definition $\mathbf{A}$ , $\mathbf{B}$ and $\mathbf{C}$ are categories, $L\$ , $X\$ , $F\$ and $M\$ are functors, and $\sigma\$ and $\alpha\$ are natural transformations.

The left Kan extension of a functor

$X: \mathbf{A} \rightarrow \mathbf{C}$

along

$F : \mathbf{A} \rightarrow \mathbf{B}$

is a pair

$(L: \mathbf{B} \rightarrow \mathbf{C}, \varepsilon : X \rightarrow LF)$

such that there is a unique

$\sigma : L \rightarrow M$

for every

$M: \mathbf{B} \rightarrow \mathbf{C}$

and every

$\alpha : X \rightarrow MF$ ,

such that the following diagram commutes.

Image:Kan extension universal property diagram.png

Where $\sigma_F (A)= \sigma (FA) \$ .

The diagram expresses the equation

$\sigma_F \circ \varepsilon = \alpha \$ .

The functor L if often written

Lan F X

[edit] Right Kan extension

In this definition $\mathbf{A}$ , $\mathbf{B}$ and $\mathbf{C}$ are categories, $R\$ , $X\$ , $F\$ and $M\$ are functors, and $\delta\$ and $\mu\$ are natural transformations.

The right Kan extension of a functor

$X : \mathbf{A} \rightarrow \mathbf{C}$

along

$F : \mathbf{A} \rightarrow \mathbf{B}$

is a pair

$(R: \mathbf{B} \rightarrow \mathbf{C}, \eta: RF \rightarrow X)$

such that there is a unique

$\delta : M \rightarrow R$

for every

$M: \mathbf{B} \rightarrow \mathbf{C}$

and every

$\mu: MF \rightarrow X$ ,

such that the following diagram commutes.

image:Right Kan extension universal property diagram.PNG

Where $\delta_F (A) = \delta (FA) \$ .

The diagram expresses the equation

$\eta \circ \delta_F = \mu \$ .

The functor R is often written

Ran F X

[edit] Properties

[edit] Existence of Kan extensions

Suppose that $X:\mathbf{A}\to\mathbf{C}$ and $F:\mathbf{A}\to\mathbf{B}$ are two functors. If the category $\mathbf{A}$ is cocomplete, then there exists a left Kan extension $Lan F X$ of $X$ along $F$ .

[edit] Kan extensions as coends

Suppose that

$K:\mathbf{M}\to\mathbf{C}$ and $T:\mathbf{M}\to\mathbf{A}$

are two functors such that for all objects m and m' of M and all objects c of C, the copowers $\mathbf{C}(Km',c)\cdot Tm$ 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,

$Lc=(\mathrm{Lan}_KT)c=\int^m\mathbf{C}(Km,c)\cdot Tm$

when the above coend exists for every object c of C.

Dually, right Kan extensions can be computed by the formula

$(\mathrm{Ran}_KT)c=\int_mTm^{\mathbf{C}(c,Km)}$ .

[edit] Limits as Kan extensions

The limit of a functor $F:C\to D$ can be expressed as a Kan extension by

lim F = Ran E F

where $E$ is the unique functor from $C$ to $1$ (the category with one object and one arrow). The colimit of $F$ can be expressed similarly by

colim F = Lan E F

[edit] References

Cartan, H., Eilenberg, S. (1956). Homological algebra. Princeton: Princeton University Press.
Mac Lane, S. (1998). Categories for the Working Mathematician. Second Edition. Springer-Verlag. ISBN 0-387-98403-8.

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Category: Adjoint functors