I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to arise on the big Zariski site.
Thanks!
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Sign up to join this communityI have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to arise on the big Zariski site.
Thanks!
Since you ask specifically for an easy example (not a natural, useful, etc., one), I suggest taking any category in which there are two objects $A$ and $B$ for which Hom$(A,B)$ is not a one-element set (i.e., the category is equivalent to neither the initial nor the terminal category) and giving it the topology in which every sieve (even the empty sieve) is covering. Then the only sheaf is the functor assigning, to each object, a singleton set, and my assumption about $A$ and $B$ ensures that the presheaf represented by $A$ is not a sheaf.
You want a topology that is not "subcanonical", the definition of which is precisely that there are representable functors that are not sheaves. You could of course take a very fine topology, for instance the discrete topology in which every sieve is a covering sieve, and then most schemes are not sheaves (if you are using the category of schemes). In fact, the only one would be the terminal object: since the empty sieve covers everything, for any sheaf object $X$ and any other object $S$, the empty map from the empty covering of $S$ extends uniquely to a map from $S$ to $X$, so $X$ is terminal.
For a less trivial answer, the canonical topology is characterized as being generated by the universal strict epimorphisms: maps $X \to Y$ such that $Y$ is the quotient of $X$ by the relations $X \times_Y X \Rightarrow X$ (two projections), and for which this is stable under base change (it's easy to see from the definition of a quotient that this makes every representable functor a sheaf). The big Zariski site won't help you because the coverings are the same as in the small site; only the objects are expanded. Any fppf covering is subcanonical, and that includes any one of the "normal" Grothendieck topologies.
I am told that Voevodsky's h-topology, or the similar cdh-topology, is not subcanonical, but I don't know anything about it.
Here is an easy, but very specific, example: let $T$ be the topology on the category of topological spaces with continuous maps where {$f_i: U_i \to X$} is a covering iff $X = \bigcup f_i(U_i)$. One can show that the only sheaves on this topology are those that are representable by topological spaces with trivial (indiscrete) topology. So for example, the presheaf represented by the discrete space $X = ${$a, b$} is not a sheaf. If $Y$ is $X$ with trivial topology, then $\{id: Y \to X\}$ is a covering, but the sequence $$\hom(X, Y) \to \hom(X, X) \rightrightarrows \hom(X, X \times_Y X)$$ is not exact.