 I use this sign when I'm talking about stuff that we're not ready for, given what the course has covered so far. If students see a sign like this, it means "Don't worry that you don't understand this stuff: you're not really supposed to. You can just skip it."

Owen wrote:

> It's bizarre to me that (certain types of) functors to \$$\mathcal{C}\$$ correspond to functors \$$\mathcal{C}\to\mathbf{Set}\$$.

That bizarreness is exactly why the Grothendieck construction is important. However, it's just a souped-up version of a construction you've probably seen, and putting it in context makes it seem less strange.

Here's the basic idea: there are two ways to think of a subset \$$S\$$ of a set \$$X\$$. One the 'inclusion' function

$i_S: S \to X$

namely the 1-1 function sending any element of \$$S\$$ to itself viewed as an element of \$$X\$$. The other is the 'characteristic function' or 'indicator function'

$\chi_S : X \to \lbrace 0,1 \rbrace$

with \$$\chi_S(x) = 1\$$ if \$$x \in S\$$ and \$$\chi_S(x) = 0 \$$ otherwise.

So, we can think of subsets of \$$X\$$ either as special maps _into_ \$$X\$$, or maps _out of_ \$$X\$$... into a special thing!

Both viewpoints are important. The first viewpoint lets us 'push forward' a subset of \$$X\$$ along a map \$$X \to Y\$$ and get a subset of \$$Y\$$. The second lets us 'pull back' a subset of \$$X\$$ along a map \$$Y \to X\$$ and get a subset of \$$Y\$$. The first method is called the 'image' (or in this course 'direct image'), while the second is called the 'inverse image'). We discussed both, and their relationship, in Chapter 1.

This idea has lots of generalizations. For example, if we replace \$$X\$$ with a category \$$\mathcal{X}\$$ and replace \$$\lbrace 0,1 \rbrace \$$ with the category \$$\mathbf{Set}\$$, we get the following fact: a specially nice kind of functor called a 'discrete fibration' \$$i_S : \mathcal{S} \to \mathcal{X}\$$ is the same as a functor \$$\chi_S : \mathcal{X} \to \mathbf{Set}\$$.

The Grothendieck construction goes even further: now we we replace \$$X\$$ with a category \$$\mathcal{X}\$$ and replace \$$\lbrace 0,1 \rbrace \$$ with the category \$$\mathbf{Cat}\$$! Now a specially nice kind of functor called 'fibration' \$$i_S : \mathcal{S} \to \mathcal{X}\$$ is the same as a functor \$$\chi_S : \mathcal{X} \to \mathbf{Cat}\$$.

It goes on, and I've written a whole big fat essay about this idea. For example, it's very important in topology. But in every case, the map I'm calling \$$i_S\$$ needs to be 'nice' in some way; it can't be arbitrary.

Despite all this stuff, I'm still not optimistic about your original question.