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 $\alpha =(\sigma K)\circ \eta$.

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 $\alpha =ϵ\circ (\sigma K)$.

A right Kan extension is pointwise iff if it is preserved by all representable functors $\mathcal{E}[e,-]$.

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:

(Ran_K F)(d) = \int_{c \in \mathcal{C}} Fc^{\mathcal{D}(d, Kc)}$$ ## Examples - Let $\mathbb \Delta$ be the [[simplex category]] of finite non-empty [[Ordinal|ordinals]] and [[Monotone Function|monotone maps]]. ## Remarks - This can be extended by analogy to any [[2-Category|2-category]] not just [[Cat (Category)|Cat]]. - Left adjoints preserve left Kan extensions. ## See also [[Pointwise Kan Extensions]] ## References - [[riehl2016-category-theory]] - [[riehl2014-categorical-homotopy-theory]] - [[maclane1978-categories]] - [[riehl2014-categorical-homotopy-theory]] - [[kan58-kan-extensions]]