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nLab > Latest Changes: sheaf

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  1.  
    • 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

    1. General definition in components

    2. General definition abstractly

    3. Characterizations over sites of opens

    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 limℱ(𝒰)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 𝒯\mathcal{T} be be a topos, regarded as a large site when equipped with the canonical topology. Then a presheaf on 𝒯\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.

    diff, v71, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 18th 2022

    Added:

    The original definition is in

    • Jean Leray, L’anneau d’homologie d’une représentation. Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946), 1366–1368. Leray-0.pdf:file

    Subsequent development by Leray, incorporating ideas of Henri Cartan:

    • Jean Leray, L’anneau spectral et l’anneau filtré d’homologie d’un espace localement compact et d’une application continue, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 29 (1950), 1–80, 81–139.

    Henri Cartan’s account of the theory:

    • Henri Cartan, Faisceaux sur un espace topologique. I, II, Séminaire Henri Cartan, Exposés 14, 15. numdam: I, II.

    It refers to a previous exposition of the theory in Exposés 12–17 of the first year (1948/1949), which apparently are not scanned, unlike Exposés 1–11.

    diff, v74, current

    • CommentRowNumber14.
    • CommentAuthorgregprice
    • CommentTimeJul 24th 2022

    Document the alternate definition where “sheaf” means “étale space”, and point to where the latter entry describes the relationship. Hopefully this reduces puzzlement for the next person who runs across that alternate definition in the wild.

    diff, v75, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2022
    • (edited Jul 24th 2022)

    Since this concerns the case of sheaves over sites of open subsets of a topological space, I have moved the paragraph up and made it a sub-section of “Characterizations over special sites”

    diff, v77, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2022
    • (edited Jul 24th 2022)

    [ accidental duplicate removed ]

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