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I have fixed that broken sentence, thanks for pointing it out.
Now I am re-arranging the Definition-section a bit. Your “quick definition” should definitely be listed among the definitions, not among the Ideas. And so I am creating more subsections now and add some glue. More later..
Okay.I have decomposed the Definintion-section at sheaf now into three subsections
To the last one I have moved your “quick definition”. I have edited it slightly for completeness. Then I have added a proof (or most of it) that it is equivalent to the previous definitions in this special case.
There is still room to fine-tune the notation, as $lim \mathcal{F}(\mathcal{U})$ is a bit of an abuse of notation. On the other hand, if you are happy with it, I am, too. :-)
By the way, now that I have looked at that MO discussion: there are also situations where a sheaf is precisely a contravariant functor that sends all small colimits to limits.
I have added a brief remark on that to Characterization over canonical topologies.
Let $\mathcal{T}$ be be a topos, regarded as a large site when equipped with the canonical topology. Then a presheaf on $\mathcal{T}$ is a sheaf precisely if it sends all colimits to limits.
Really? I thought this was one of the advantages that you get by moving to the $\infty$-world that isn’t true in the 1-world. Where can I find a proof of that?
Since every sheaf on the canonical topology of a Grothendieck topos is representable, sheaves certainly send all colimits to limits. On the other hand, if a presheaf sends all colimits to limits, then by the special adjoint functor theorem it has a left adjoint and is therefore representable.
Ahm, sorry, maybe I am being mixed up. I was thinking this follows directly from all epis being regular.
[This message overlapped with Zhen Lin’s.]
@Urs: I don’t immediately see how a functor preserving regular epis implies that it preserves coequalizers, unless it also preserves finite limits.
@Zhen: Sure, but what about large-set-valued presheaves? That’s what I understood the theorem to be about. If it’s only about the small-set-valued ones, then it should say so explicitly.
I don’t immediately see how a functor preserving regular epis implies that it preserves coequalizers, unless it also preserves finite limits.
That’s a good point which ought to be recorded. Does anyone know of a good example of this?
I was thinking of small-set valued presheaves. I assumed we know that every canonical sheaf is representable. Then it remains to see that every functor sending colimits to limits is a sheaf. But such a functor preserves the coequalizer that gives the regular epi and hence is a sheaf.
That was my reasoning at least. I added a paranthetical remark concerning the small sets in the codomain.
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