Topologies

OK, this will be another quick one.

We want to introduce the notion of a Grothendieck topology on a category $\mathsf{C}$.
The goal of this very general notion is to be able to speak about sheaves on $\mathsf{C}$. Let’s recall a few things.

Let’s fix a topological space X.

A presheaf $\mathscr{F}$ on X is the data of a set $\mathscr{F}(U)$ for each open subset $U \subseteq X$, together with a map of sets $\mathscr{F}(U) \to \mathscr{F}(V)$ for all inclusions $V \subseteq U$ (called restrictions), satisfying the obvious compatibilities.

Explicitly the compatibilities are as follows: given inclusions $W \subseteq V \subseteq U$ then we have two maps $\mathscr{F}(U) \to \mathscr{F}(V) \to \mathscr{F}(W)$ and $\mathscr{F}(U) \to \mathscr{F}(W)$ (the latter coming from the inclusion $W \subseteq U$) and we ask they be the same. Elements $s \in \mathscr{F}(U)$ are called sections of $\mathscr{F}$ over U. We denote $s|V$ the image of s in $\mathscr{F}(V)$, under the appropriate restriction map.

There is a more compact way to describe what a presheaf is. Define the category $\mathsf{Op}X$ whose objects are the open subsets of X and where the hom-set between V and U is empty if $V \nsubseteq U$ and is a singleton if $V \subseteq U$.

A presheaf on X is just a contravariant functor from $\mathsf{Op}X$ to $\mathsf{Set}$

We can see that in this definition the topology of X hasn’t really played a huge role. It does play a role in the definition of a sheaf. Essentially, a sheaf is a presheaf where global data is glued from local data. That is, if $\{U_i\}$ is an open cover of U, then $\mathscr{F}(U)$ is obtained by gluing the $\mathscr{F}(U_i)$. More precisely:

A presheaf is separated if given two sections $s,t \in \mathscr{F}(U)$ and a covering $\{U_i\}$ of U such that $s|U_i = t|U_i$ for all i, then s=t.

A presheaf satisfies gluing if given a collection of sections $s_i \in \mathscr{F}(U_i)$ such that $s_i|U_i \cap U_j = s_j | U_j \cap U_i$ then there exists a section s over U such that $s|U_i = s_i$.

Examples of sheaves and presheaves abound in nature. For example we have a sheaf of continuous functions from X to the real numbers, or smooth functions if X is a smooth manifold (or holomorphic if X is complex). An example of presheaf which is not always a sheaf is the presheaf of bounded functions from a topological space X into the real numbers. This presheaf is separated but doesn’t satisfy gluing in general.

Again there is a more compact way to state when a presheaf is a sheaf.
Again let $\mathscr{F}$ be a presheaf, U be an open subset of X and $U_i$ a covering of U.
We have a map from $\mathscr{F}(U) \to \prod_i\mathscr{F}(U_i)$ sending a section s to the sequence of restrictions $(s|U_i)_i$.
We also have another two maps $\prod_i\mathscr{F}(U_i) \rightrightarrows \prod_{j,k}\mathscr{F}(U_{ij})$ (where $U_{ij} = U_i \cap U_j$) defined by restricting on the first and second component respectively.

A presheaf is a sheaf if, in the above context, the following sequence is exact $\mathscr{F}(U) \to \prod_i\mathscr{F}(U_i) \rightrightarrows \prod_{j,k} \mathscr{F}(U_{ij})$

(that is, the guy on the left is the equaliser of the guys on the right)

What should transpire by the discussion above is that if we are only interested in sheaves on X, we only need to know about coverings of X. This idea is the key to generalise topologies to arbitrary categories.

Given an object U of a category C, a topology on C will tell us that some collections of arrows $\{U_i \to U\}$ are coverings of U.
Before we give the exact definition we should think of what should one use to replace intersections.

If $U_i,U_j \subseteq U$ then it is trivial to see that $U_{ij}$ is the fibre product of $U_i$ and $U_j$ over U. This is the correct replacement for the notion of intersection.

A Grohendieck topology on a category C is a collection Cov U of coverings for each object U, satisfying some axioms. That is, Cov U is a collection of families of morphisms $\{U_i \to U\}$ which the topology declares to be coverings. The axioms they satisfy are as follows

• $\{id: U \to U\} \in Cov U$, the family consisting of just the identity of U is a covering of U
• If $\{U_i \to U\} \in Cov U$ and $\{ V^i_j \to U_i\} \in Cov U_i$ then $\{V^i_j \to U_i \to U\} \in Cov U$
• If $\{U_i \to U\} \in Cov U$ and $V \to U$ is any morphism in C then $\{U_i \times_U V \to V\} \in Cov V$

The first axiom is saying the the covering consisting of the whole space is a covering; the second states that if you start with a covering of U and then cover each open of that covering with other open sets, then the latter forms a covering of U; the third axiom says that if you start from a covering of U and V is any open subset of U then we can restrict (i.e. intersect) to obtain a covering of V.

A site is a category endowed with a grothendieck topology.

Now we want to define sheaves on a site C. A presheaf will simply be a functor $\mathscr{F}: \mathsf{C}^\circ \to \mathsf{Set}$. A sheaf is a presheaf satisfying the exactness condition we’ve seen above, where now U varies through all objects of C and the collection $\{U_i \to U\}$ varies through all coverings of U (wrt the topology of C).

Now, I’m sure that if you’ve never seen all this before, you’re quite confused. We’ve rushed a lot. So, we’ll rush even more by saying a few words about limits and colimits in the category $\mathsf{Sh C}$ of sheaves (of sets) on a site C.
The Yoneda embedding $\mathsf{C} \to \mathsf{PSh C}$ preserves limits, therefore things like fibre products remain intact when transported inside the category of presheaves.
Both limits and colimits in PSh C are computed pointwise. That is, a (co)limit $(co)lim_i F_i$ is a presheaf, whose value at an object U is given by the corresponding (co)limit in Set: $((co)lim_i F_i)(U) = (co)lim_i (F_i(U))$.
There is a forgetful functor $ShC \to PSh C$, which admits a left adjoint called sheafification. In our case, sheaves with values in Set, it’s easy to construct it (set-theoretic difficulties aside), no pun intended.
Hence limits in ShC are computed pointwise as well, and once again fibre products remain intact!
For colimits it’s a bit more complicated, but not tremendously so: the colimit of a diagram of sheaves will just be the sheafification of the colimit of the same diagram as presheaves. (someone please correct if I’m wrong!)

Anyways, this ought to be enough. Next time, we’ll finally have a look at some algebraic geometry.

(for the pedants, all categories above are assumed to have fibre products)