Owen wrote:

> I'm excited about Kan extensions! I've come across them before but I've never been able to make a gut connection with them on the level of adjoints and limits, which they apparently generalize.

They're a lot less complicated than most people manage to make them seem. They're a special case of adjoint functors.

Here's an ultra-terse explanation which may only make sense to me; I'll expand it later
in this class. We've seen here that for any categories \$$\mathcal{C}\$$ and \$$\mathcal{D}\$$, composing with \$$G : \mathcal{D} \to \mathcal{C}\$$ turns functors \$$\mathcal{C} \to \mathbf{Set}\$$ into functors \$$\mathcal{D} \to \mathbf{Set}\$$. We'd like to 'reverse' this process. Here's how. This process is actually a _functor_ - so to 'reverse' it, we take its left or right adjoint! Its left adjoint is called 'left Kan extension', and its right adjoint is called 'right Kan extension'.

In a bit more detail:

There's a _category_ of functors \$$\mathcal{C} \to \mathbf{Set}\$$, called \$$\mathbf{Set}^\mathcal{C}\$$ . There's a _category_ of functors \$$\mathcal{D} \to \mathbf{Set}\$$, called \$$\mathbf{Set}^\mathcal{D}\$$ . Composing with \$$G : \mathcal{D} \to \mathcal{C}\$$ gives a functor from the first of these categories to the second:

$\textrm{composition with } G : \mathbf{Set}^\mathcal{C} \to \mathbf{Set}^\mathcal{D}$

This functor has left and right adjoints, which are called **left Kan extension** and **right Kan extension**.