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• CommentRowNumber1.
• CommentAuthorAaron F
• CommentTimeOct 4th 2012
I recently added a quick definition of sheaf, found in a MathOverflow comment, to the article on sheaves. I've convinced myself that it's accurate, but I'm afraid I may have missed something, so I would appreciate some independent verification. I'm also not sure where in the article this short definition should go.

I also believe I may have found a typo in the definition that was there before. Unfortunately, I don’t understand the definition well enough to know what’s wrong (if anything).
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 4th 2012
• (edited Oct 4th 2012)

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..

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 4th 2012
• (edited Oct 4th 2012)

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. :-)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 4th 2012
• (edited Oct 4th 2012)

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.

• CommentRowNumber5.
• CommentAuthorAaron F
• CommentTimeOct 11th 2012
Whoa! I go away for a week, and come back to find that someone has done all my work for me. :D The article looks great now---many thanks.

I'd never seen the definition of limits in terms of products and equalizers before, so I'm glad you mentioned it! I always thought that construction was sort of an ad-hoc trick, so it's nice to see it explained in a general way.
• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 11th 2012

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?

• CommentRowNumber7.
• CommentAuthorZhen Lin
• CommentTimeOct 11th 2012

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 11th 2012
• (edited Oct 11th 2012)

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.]

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeOct 11th 2012

@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.

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeOct 11th 2012

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?

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 11th 2012

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.

• CommentRowNumber12.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 20th 2021

Added the original paper by Leray.