User:Robertbyrne/Kan extension

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."

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.

$\sigma :L\to M$,

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.

$\delta :M\to R$,

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

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 $X:\mathbf{𝐀}\to \mathbf{𝐂}$ along $F:\mathbf{𝐀}\to \mathbf{𝐁}$ is a pair $(L:\mathbf{𝐁}\to \mathbf{𝐂},ϵ:X\to LF)$ such that there is a unique $\sigma :L\to M$ for every $M:\mathbf{𝐁}\to \mathbf{𝐂}$ and $\alpha :X\to MF$, such that the following diagram commutes.

File:Kan extension universal property diagram.png

Where ${\sigma }_F(A)=\sigma (FA)$.

The diagram expresses the equation ${\sigma }_F\circ ϵ=\alpha$.

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 $X:\mathbf{𝐀}\to \mathbf{𝐂}$ along $F:\mathbf{𝐀}\to \mathbf{𝐁}$ is a pair $(R:\mathbf{𝐁}\to \mathbf{𝐂},\eta :RF\to X)$ such that there is a unique $\delta :M\to R$ for every $M:\mathbf{𝐁}\to \mathbf{𝐂}$ and $\mu :MF\to X$, such that the following diagram commutes.

File:Right Kan extension universal property diagram.PNG

Where ${\delta }_F(A)=\delta (FA)$.

The diagram expresses the equation $\eta \circ {\delta }_F=\mu$.

End formula

Left Kan extension

The object function of the left Kan extension, $L$, as defined earlier, is:

$L(B)={\int }^A\mathbf{𝐁}(FA,B)\cdot XA$.

We note that when this coend exists we have the object function of the left Kan extension, however, we don't have that the existence of the left Kan extension implies the existence of this coend.

The unit, $ϵ$, of the left Kan extension is defined:

${ϵ}_A={\omega }_{A,FA}\circ i_{1_{FA}}$, where $\omega$ is the ending wedge of the coend above, and $i$ is the injection from $X$ into the copower

$\mathbf{𝐁}(FA,FA)\cdot XA$.

Right Kan extension

The object function of the right Kan extension, $R$, as defined earlier, is:

$R(B)=\underset{A}{\int }(XA{)}^{\mathbf{𝐁}(B,FA)}$.

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.