Right, it’s nearly Christmas and I have been so busy working on my undergrad thesis that I haven’t had any time to fire off another proper post. So I just wanted to record a few more basic facts about Grothendieck topologies further to my last post about them!

Let $\mathcal{C}$ be any small category. Just as for an arbitrary topological space, there are two obvious (and not always so interesting) Grothendieck topologies we can put on $\mathcal{C}$:

The Indiscrete Topology: for each $U\in\mathcal{C}$, let $J_{\text{ind}}(U)$ consist of just the maximal sieve $\bigcup_{A\in\mathcal{C}} \mathcal{C}(A,U)$. This is analogous to the indiscrete topology on a topological space where the only open sets are the empty set and the whole set. The Grothendieck pretopology associated to this topology (if $\mathcal{C}$ has pullbacks) has only isomorphisms as covering families, so it’s a strange topology in which you can’t cover “open sets” with any smaller or larger “open sets”.

Another noticeable thing about the indiscrete topology $J_{\text{ind}}$ is that every presheaf $F$ on the site $(\mathcal{C}, J_{\text{ind}})$ is a sheaf; this is because the only sieve on an object $U$ is the maximal sieve, which can be identified with the respresentable presheaf $h_U = \mathcal{C}(-,U)$, and every morphism $h_U \rightarrow F$ in $[\mathcal{C}^{op}, \text{Set}]$ has exactly one extension to a morphism $h_U \rightarrow F$; namely, the identity natural transformation.

This is analogous to equipping a space with the usual indiscrete topology; the sheaf axioms are automatically satisfied for any presheaf because the only open sets to worry about are the whole space and the empty set. There is not enough “separation” between points to mess up a presheaf becoming a sheaf.

The discrete topology: Conversely, the “finest” possible topology on a category is the discrete topology $J_{\text{dis}}$ in which every sieve is a covering sieve. Here every single covering family is a covering of a set. In contrast to the indiscrete topology, the only presheaf on $(\mathcal{C}, J_{\text{dis}})$ that is a sheaf is the terminal object in the category $[\mathcal{C}^{op}, \text{Set}]$ i.e. the presheaf sending every object to a one-point set. There is just “too much separation” going on for sheaves to behave nicely.

Apart from these two extremes, there is also another checkpoint in the lattice of Grothendieck topologies on a category; this is called the canonical topology. As we have seen, the more covering sieves are introduced into a topology, the fewer sheaves are left on the category. There are certain presheaves that, often, we would like to be sheaves; these are the representable presheaves $h_U = \mathcal{C}(-,U)$. As we’ll later see, schemes turn out to be certain sheaves on the “Zariski site”, and these are closely linked to the concept of representability.

The canonical topology is the largest Grothendieck topology on a category on which every representable presheaf $h_U$ is a sheaf. Any topology where every representable presheaf is a sheaf is called subcanonical, and in practice most useful topologies are subcanonical – the Zariski topology, étale topology, fppf and fpqc topologies, …

One day in the future we’ll get to work with these exotic things but at the moment we’ll stick with the first. Anyway, I’ll get back to this sometime soon; Merry Christmas and a happy New year!