Kan Extension

Idea

A Kan extension provides a method for extending a functor along another functor. It generalizes the concept of extending a function from a subspace to a larger space by finding the best possible approximation that factors through a given morphism.

Definition

Let $F:\mathcal{C}\Rightarrow \mathcal{E}$ and $K:\mathcal{C}\Rightarrow \mathcal{D}$ be functors.

Left Kan Extension

The left Kan extension of $F$ along $K$ is a functor $La{n}_{K}F:\mathcal{D}\Rightarrow \mathcal{E}$ equipped with a natural transformation $\eta :F\Rrightarrow (La{n}_{K}F)\circ K$ such that for any functor $G:\mathcal{D}\Rightarrow \mathcal{E}$ and natural transformation $\alpha :F\Rrightarrow G\circ K$, there exists a unique natural transformation $\sigma :La{n}_{K}F\Rrightarrow G$ such that $\alpha =(\sigma K)\circ \eta$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C} \arrow[rr, "K"] \arrow[rd, "F"'] & & \mathcal{D} \arrow[ld, "Lan_K F"] \\
& \mathcal{E}
\end{tikzcd}
\end{document}

Right Kan Extension

The right Kan extension of $F$ along $K$ is a functor $Ra{n}_{K}F:\mathcal{D}\Rightarrow \mathcal{E}$ equipped with a natural transformation $ϵ:(Ra{n}_{K}F)\circ K\Rrightarrow F$ such that for any functor $G:\mathcal{D}\Rightarrow \mathcal{E}$ and natural transformation $\alpha :G\circ K\Rrightarrow F$, there exists a unique natural transformation $\sigma :G\Rrightarrow Ra{n}_{K}F$ such that $\alpha =ϵ\circ (\sigma K)$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C} \arrow[rr, "K"] \arrow[rd, "F"'] & & \mathcal{D} \arrow[ld, "Ran_K F"] \\
& \mathcal{E}
\end{tikzcd}
\end{document}

Adjunction Perspective

Kan extensions are characterized as adjoints to the precomposition functor $K^{\ast }:[\mathcal{D},\mathcal{E}]\Rightarrow [\mathcal{C},\mathcal{E}]$ defined by $K^{\ast }(G)=G\circ K$. When they exist for all $F$, we have a sequence of adjoints: $La{n}_K⊣K^{\ast }⊣Ra{n}_K$ This yields the natural isomorphisms: $Nat(La{n}_{K}F,G)\cong Nat(F,G\circ K)$ $Nat(G\circ K,F)\cong Nat(G,Ra{n}_{K}F)$

Formulae via (Co)ends

If $\mathcal{C}$ is small and $\mathcal{E}$ is cocomplete, the left Kan extension exists and can be computed pointwise using an end:

$$(La{n}_{K}F)(d)={\int }^{c\in \mathcal{C}}\mathcal{D}(Kc,d)\cdot Fc$$

Dually, if $\mathcal{E}$ is complete, the right Kan extension exists and is computed using a coend:

$$(Ra{n}_{K}F)(d)={\int }_{c\in \mathcal{C}}F{c}^{\mathcal{D}(d,Kc)}$$

References

• riehl2016-category-theory
• maclane1978-categories
• riehl2014-categorical-homotopy-theory
• kan58-kan-extensions