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See this MO question. I got the answer wrong, because I forgot the definition of sieves generated by a Grothendieck pretopology. However, the sort of thing in the question I use in my thesis (albeit the singleton version), so now I curious as to how this works. The following is my thought process, please tell me if it is wrong.
If J is a pretopology on a category $C$ with pullbacks, then let $J$-epi denote the singleton pretopology with covers maps that admit local sections over a $J$-cover. Now if $C$ is extensive, $J$-epi is the same as $\amalg J$-epi, where $\amalg J$ is a class of maps $\amalg U_i \to X$ for $\{ U_i \to X\}$ a $J$-covering family. If $J$ isn’t a superextensive pretopology, then $J$-sheaves are not necessarily the same as $\amalg J$-sheaves, but sheaves for $J$-epi can’t be sheaves for both $J$ and $\amalg J$, so we have a pretopology $K$ on $C$ countering the claim that $K$-sheaves are the same as $K$-epi sheaves.
However, the question relates not to $J$-epi, but to a non-singleton version: the new covering families are such that old covering families factor through them. We can think about just the sheaf condition for a single object $X$. I could take a new covering family of $X$ to be $\{ U_i \amalg Y \to X\}$ for any old map $Y \to X$, and hardly expect the original sheaves to be sheaves for this covering family.
Incidentally, in the reference - Goodwillie and Lichtenbaum’s paper on the h-topology - the OP (what on earth does that mean??) gave, it mentions the f-topology on schemes as being the smallest topology generated by the extensive topology and the topology of finite surjections. Another good example for a superextensive (pre)topology. G&L cite SGA4 for their definition of ’topology’ and that the question has an obvious answer ’yes’. I don’t have access to SGA4 at the moment, so I can’t check the precise definition. It may be hidden in there.
Also G&L mention a condition of a topology having enough points: that a sheaf if trivial iff all its stalks are trivial. Do we have something like this on the Lab? I really need to get back to work, else I’d dig around myself..
OP = Original Poster. I added an answer to the MO question; despite what one might “hardly expect,” it is actually true. (-:
Cool, thanks!
I suppose the obvious thing to ask is whether this is true for sites without all pullbacks, and replacing local-section-admitting maps (or the non-singleton version as in the question) with only those of which all pullbacks exist.
The version in the Elephant is stated for categories without pullbacks. The right thing to do is still talk about local-section-admitting maps, just phrase it without using pullbacks.
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