Grothendieck topology and relation with usual topologies

Question

Recently I stumbled upon the definition of $\textbf{Grothendieck}$ $\textbf{topologies}$ of a category $\mathcal{C}$. I do know that is one of the most interesting parts of the contemporary algebraic approach for topology and geometry as well. Though, I was curious to understand the aim of this particular name and the correlation with the usual notion of the topology on a usual set; if for instance is a generalization or the usual topology is a kind of a special restriction of the former definition. Although I did find some interesting articles about it (for instance https://ncatlab.org/nlab/show/historical+note+on+Grothendieck+topology) I didn't understand exactly.

So, can we recover the usual definition of a topology on a set $X$ from the definition of a $\textbf{Grothendieck}$ $\textbf{topologies}$ on a specific category $\mathcal{C}$? If not, what can we define through that definition with an analogue in the usual point-set topology?

Answer 1

I think the wikipedia page is actually pretty good. Here's a super-quick summary, though:

• To a topological space $X$, we can associate a category $Open(X)$ whose objects are the open sets of $X$, and whose morphisms are the inclusion maps (this is discussed here). While in general $X$ cannot be recovered from $Open(X)$ (think about indiscrete topological spaces), if $X$ is sober then it can be; and the vast majority of naturally-occurring topological spaces are sober. So, for most intents and purposes you can conflate $X$ with $Open(X)$.

• A sheaf on $X$ is then a contravariant functor from $Open(X)$ satisfying some properties - specifically, locality and gluing. The one you want to focus on here is the gluing property, for which we need the notion of a family of open sets covering another open set.

• A Grothendieck topology is basically what you get when you ask for a category which behaves like the category of open sets in the sense that it has a good notion of covering. What do I mean by this? Well, given a category $C$, we want to think of elements of $C$ as being open sets in some imaginary topological space $X_C$ (which might not, in fact, exist); the Grothendieck topology just tells us when some family of "opens" covers some other "open".

• What exactly does this mean? Well, a first guess would be that we want a set $\mathcal{S}$ of pairs $(a, M)$ where $a$ is an object of $C$ and $M$ is a family of morphisms with codomain $a$ (where $(a, M)\in\mathcal{S}$ should mean "$M$ covers $a$"). It turns out that this isn't quite right, and what we actually want is more complicated - this is the notion of a sieve.

• So basically, a site (= a category with a Grothendieck topology) is a context for sheaf theory. Note that while every topological space yields a site, there are some sites which do not come from any space (see ???). The definition of a sheaf on a site is slightly complicated, but the wikipedia page does a decent job of explaining it here.

• Why would you want a notion of sheaf theory for objects more general than topological spaces? Well, the original motivation (to my understanding) was to develop a notion of etale cohomology for schemes; so if you care about schemes, you should care about sites. Since then, I know that Grothendieck topologies have also been used to give a categorial interpretation of forcing (see e.g. Maclane and Moerdijk's book, "Sheaves in geometry and logic").