In all honesty, I’m not going to blog all the gory details of all the definitions of things that go into the definition of a stack. This leaves me with a problem I haven’t quite figured out how to solve: What do I show? The task seems so massive right now, but there must be a non-offensive way to sort of give the idea of a stack by being more precise than the general vague way people try to explain it, but without doing more than say 5 more posts on it (I also would like to avoid cheating, i.e. go through descent rather than just saying “a stack is a sheaf on a site”).
So today we’ll jump back a little and try to figure out why one would care about stacks. Recall the Yoneda Lemma. Colloquially this says that if we have a (small) category , then the functor of points, which is by (note for us it’s contravariant) completely determines the object . So what is happening is that we are “testing” what the object looks like, and if we test it against everything, then we completely know what the object looks like.
If we’re thinking about differential geometry, then by functor of points, we really mean “points”. Let be a smooth manifold in the category of smooth manifolds. Then we’ll test what the point-valued points look like. Well, apply the functor of points to a point, . Each map is completely determined by which point of the point goes to. It sounds silly, but this is just saying “the points of are the point-valued points”, which we already knew.
Let’s think about this from a very classical algebraic geometry standpoint. Let’s work in the category of affine schemes. We call the -valued points of . This because if we have something like , then by -valued points we really mean what are the solutions to when can take values in . So if we think of it is the complex curve , or are the integer solutions (only trivial ones by Fermat’s Last Theorem). So scheme theoretically this tool is really cool. It groups all of this information into one package.
In reality, we actually usually think of Yoneda in terms of the embedding it gives us. By identifying with the object (i.e. the functor represents in the usual sense of a representable functor) we get an embedding by the obvious thing . So our category is actually sitting inside the category of functors from to .
To quickly recap in the language of schemes, given a scheme , we get a functor. But the other way around is not always true, i.e. not every functor is represented by a scheme. Here is where the notion of a stack (or algebraic space) might be nice. Sometimes it is really easy to write down what the functor of points should be for something. Now the question becomes, is this a scheme? Or maybe a better question should be does it matter if it is or is not a scheme? Exactly how loose do we allow the definition to get before we shouldn’t consider it as some sort of “geometric space”?
Just a last quick example to show this really does happen. Suppose you want to figure out what the space of curves of genus looks like. Call this space . Then you can almost immediately write down that the functor of points for this space needs to be something like where is proper, smooth, and the fibers are curves of genus and we mod out by appropriate isomorphisms (maybe if we think about moduli later we’ll be more careful with that definition). This turns out to NOT correspond to a scheme. But it is still quite nice, and is in fact a stack. Moral of the story: If we think of schemes as a certain class of functors, it is very natural to come to a more general version of schemes.