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 ) 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?
$\endgroup$2 Answers
$\begingroup$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").
EDIT: I'm no longer convinced this answer is worthwhile. The analogy should be between sheaves and open sets, not sheaves and closed sets. I'm leaving this answer up because it's still not completely worthless - and everything it says is still true.
One way to think about topologies is as monads on powersets. To be precise: let $\mathbb 2$ be the category $0\!\rightarrow\!1$ and let $X$ be a set, which we can also think of as a discrete category. Then $\mathbb 2^X$ (the category of functors from $X$ to $\mathbb 2$) is the powerset of $X$, made into a category by the inclusion ordering of subsets.
The monads $m:\mathbb 2^X\rightarrow \mathbb 2^X$ give "closure operators" on the subsets of $X$. More precisely they are the operators that for every subset $S\subseteq X$ obey $S\subseteq m(S)$ and $m(m(S))= m(S)$. Given a topology on $X$ we have a closure operator $\mathrm{cl}$ taking a subset $S$ to its topological closure $\mathrm{cl}(S)$. The operator $\mathrm{cl}$ has the additional property that it preserves finite intersections; $\mathrm{cl}(S\cap R)=\mathrm{cl}(S)\cap\mathrm{cl}(R)$. Categorically, preserving intersections in a partial order is the same as preserving categorical products.
In fact this gives an alternative definition of topology:
The topologies on $X$ are in bijective correspondence with monads on $\mathbb 2^X$ that preserve finite products.
Now it turns out that there is a relation between topologies and Grothendieck topologies. In particular because Grothendieck topologies can be characterised by an analogous statement:
The Grothendieck topologies on $\mathcal C$ are in bijective correspondence with idempotent monads on $\mathrm{Set}^{\mathcal{C}^\mathrm{op}}$ that preserve finite limits.
To see how this corresponds to the usual definition, note that the monad here is the sheafification functor that turns a presheaf ("element of $\mathrm{Set}^{\mathcal{C}^\mathrm{op}}$") into a sheaf.
There are three points of the above statement that we couldn't have guessed just by analogy:
- Since discrete categories are equal to their opposites we had no way of guessing if we would get $\mathrm{Set}^{\mathcal{C}}$ or $\mathrm{Set}^{\mathcal{C}^\mathrm{op}}$.
- Since every monad on a preorder is idempotent we had no way of guessing if this was a required property or just a coincidence.
- Since every limit in a preorder is a product we had no way of guessing if the condition was going to be "preserves finite products" or "preserves finite limits".
Okay. So now lets see if we can use this analogy to turn a topology on $X$ into a Grothendieck topology on $X$ considered as a discrete category (in fact we will get very close but then fail for an interesting reason). We have a monad on $\mathbb 2^X$ and we want to turn it into a monad on $\mathrm{Set}^X$ (no need to write the $\mathrm{op}$, since $X$ is discrete). Our weapon will be the adjunction between $\mathbb 2$ and $\mathrm{Set}$. The right adjoint $R:\mathbb 2\rightarrow\mathrm{Set}$ sends $0$ and $1$ to sets with $0$ and $1$ elements. The left adjoint $L:\mathrm{Set}\rightarrow\mathbb 2$ sends the empty set to $0$ and all nonempty sets to $1$. We then also get an adjunction $(R',L')$ between $\mathbb 2^X$ and $\mathrm{Set}^X$, by composing by $R$ and $L$.
So then given a monad $\mathrm{cl}$ on $\mathbb 2^X$ we can compose with the adjunction to get a monad $R'\circ\mathrm{cl}\circ L'$ on $\mathrm{Set}^X$. We can describe this monad completely. An object of $\mathrm{Set}^X$ is just a set for each element of $X$. The monad looks at the subset of $X$ on which these sets are nonempty, takes its closure with $\mathrm{cl}$ and then returns the object of $\mathrm{Set}^X$ which gives a singleton set for each element in the closure and the empty set otherwise.
So does this give a Grothendieck topology on $X$? Well it's easy to check that it's a monad, and it's clearly idempotent. In fact it also preserves finite products. But sadly it doesn't preserve all finite limits, because it doesn't preserve equalisers. This failure can be traced back to the fact that the functor $L$ doesn't preserve equalisers (consider the equaliser of the two maps from the one element set to the two element set).
So in summary:
$\endgroup$Topologies on $X$ correspond to Grothendieck topologies on $X$ considered as a discrete category, except for the fact that the sheafification functor fails to preserve equalisers.
