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• CommentRowNumber1.
• CommentAuthorDavidRoberts
• CommentTimeOct 29th 2010

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.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeOct 29th 2010

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.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeOct 29th 2010

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..

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeOct 29th 2010

OP = Original Poster. I added an answer to the MO question; despite what one might “hardly expect,” it is actually true. (-:

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeOct 29th 2010

Cool, thanks!

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeOct 29th 2010

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.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeOct 29th 2010

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.