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**.