Topology. It’s something that every math student has to become comfortable with. First you get to learn metric topology (usually on $\mathbb{R}^1$), followed by point set topology. Then in graduate school, pretty much everyone gets an introduction to algebraic topology. However, it all really just studies sets with distinguished collections of subsets. There is, however, a more general version of topology, in fact, a way to put a topology onto a category due to, of course, Grothendieck.

So the first question is: how should topologies be defined? The traditional way is that, given a set $X$, a topology $\tau$ is a collection of subsets satisfying the following properties:

1. $\emptyset, X\in\tau$

2. If $U_\alpha\in \tau$ with $\alpha$ an arbitrary index set, then $\bigcup_\alpha U_\alpha\in \tau$

3. If $U,V\in \tau$ then $U\cap V\in \tau$.

There are all sorts of other ways to define a topology, but that’s the standard one. It works out rather nicely, as any topologist can tell you, but somehow, it doesn’t satisfy algebraic geometers. Why, you might ask, would this be inadequate? The simple reason is that in algebraic geometry, there are a lot of situations where there isn’t a “good” topology to put on the set in question.

When algebraic geometers work over the complex numbers, they get two topologies to work with, the classical topology, which has the small open sets that analysts and differential geometers love so much. Also, though, there is the Zariski topology, which has the property that every open set is dense. It is beloved by algebraists for the simple reason that it captures a lot of the algebraic information. For instance, dimension can be defined by using irreducible subsets in the Zariski topology, which just correspond to irreducible polynomials. There’s a very strong algebra-geometry dictionary for this topology.

Unfortunately, when algebraic geometers decide to work in positive characteristic, say in the algebraic closure of $\mathbb{Z}/p\mathbb{Z}$ for some prime number $p$, there’s only the Zariski topology. So we need some way to generalize the notion of topology so that all varieties have “small” open sets.

Now we return to the math. Let $X$ be a topological space. Then we can define a category whose objects are the open sets in $X$ and such that, for $U,V$ open, $\mathrm{hom}(U,V)$ consists of a single element if $U\subset V$ and is empty otherwise. We’ll call this $\mathbf{Cat}_X$, and we can construct this for any topological space.

Grothendieck came up with the brilliant insight that the important things are covers. He looked at coverings and came to the conclusion that covers also had three important properties, just like topologies do:

1. If $U$ is an open set, then $\{U\subset U\}$ is a cover, that is, open sets cover themselves.

2. Coverings pull back, which means that if $\{V_i\subset U\}$ is a covering and $W\subset U$, then $\{(V_i\cap W)\subset W\}$ is a covering.

3. Coverings compose, that is, if $\{W_{ij}\subset V_i\}$ and $\{V_i\subset U\}$ are all covers, then $\{W_{ij}\subset U\}$ is a cover.

If you think about it, these are all categorical notions. The first one sounds a lot like the identity morphism, the second and third are even named suggestively as pullback and composition. So let’s move on to what Grothendieck suggested:

A Grothendieck Topology (note, the wikipedia article defines Grothendieck Topologies differently than I do) on a category is a collection of sets of morphisms called coverings such that

1. Any isomorphism is a covering

2. If $\{V_i\to U\}$ is a covering and $W\to U$ any map, then $V_i\times_U W$ exists for each $i$ and $\{V_i\times_U W\to W\}$ is a covering

3. If $\{W_{ij}\to V_i\}$ is a covering and $\{V_i\to U\}$ is a covering, then $\{W_{ij}\to U\}$ is a covering.

We call a category with a Grothendieck topology a site.

The only thing here that requires an explanation is the second condition. We should think of that fiber product as an intersection. Look at sets, and take two inclusions of sets into a third. The pullback is then going to turn out to be intersection, because it is the largest object that injects into each of them and makes all the maps commute.

So we should think of this as a generalization of a topology by first taking the coverings as fundamental, and then looking at what it means in a category. So why does this help with the loss of small open sets in algebraic geometry? Because we always have MANY natural sites to choose from, given a variety. I’ll define a few:

1. The Small Zariski Site has objects the open immersions of varieties into $X$, with morphisms the maps that commute with immersion. The coverings are the collections of morphisms whose image covers $X$, this gives something that can be thought of as the standard Zariski topology.

2. The Big Zariski Site has as objects varieties over $X$, that is, varieties equipped with a morphism to $X$ with the covers being collections of open immersions that cover the target.

3. The Small Etale Site has objects varieties over $X$ where the structure map is etale, which just means smooth of relative dimension zero. The coverings are then collections of arrows $\{Y_i\to Z\}$ such that the arrows are etale and commute with the structure maps and that the images cover $Z$.

4. The Big Etale Site is much like the Big Zariski Site, except that the maps must all be etale.

5. The fppf Site is similar to the Big Zariski and Big Etale sites, as it is on the category of schemes over $X$ with structure and covering maps all flat and of locally finite presentation. (fppf is from the French for this set of conditions.)

That’s plenty for now, at some point I’ll come back and define sheaves on sites, descent, and perhaps get to stacks, but that will have to wait until I understand them. I will mostly be following the lecture notes from the MSRI Deformation Theory and Moduli Spaces workshop, with additional references brought in, because that’s where I first got rigorous definitions of these things.