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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2010
    • (edited Oct 16th 2012)

    polished category of sheaves slightly

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 7th 2010
    I think I'll add a little bit of information about the induced topology from SGA 4.1.ii.5 as well as some explanation of some of the terminology over at scheme that directly deals with the category of sheaves.
    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeFeb 7th 2010

    I just posted a comment on cafe about Tom Leinster's remarks. Urs's comment parallel to the above entry does not have to do with etale space realization of sheaves nor more general adjunction between relative spaces and sheaves of sets; it is about another adjunction, just remotely related. Etale point of view does not work for Grpthendieck topologies and it does not work for sheaves with values in some other categories. Van Osdol etc. have studied for which categories sheafification exists; this is all different direction of generalizations.

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 7th 2010
    @Zoran: Question for you. Is there any way to define an open immersion for a general category of sheaves based on the underlying site?
    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeFeb 7th 2010

    I am not sure what you want to accomplish, I mean open immersion has a rather special meaning when working over Aff.

    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 8th 2010
    • (edited Feb 8th 2010)
    Apparently Kontsevich and Rosenberg have defined it in enough generality for my purposes. I'm reading the paper now.

    http://www.mpim-bonn.mpg.de/preprints/send?bid=2331
    if you're interested.

    Edit: Actually, I think you may have said something about this paper in the thread on covering spaces.
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2011

    I have worked on the entry category of sheaves.

    • I have reorganized the existing material a bit in an attempt to make it more systmatic.

    • I have added some more little details like more pointers to references.

    • I have addedd a new subsection of dense sub-sites. I think I will split off that material now in a separate entry.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2011

    At category of sheaves I have started a section Proofs supposed to contain details on the statements made elsewhere in this entry (this is for some students).

    Maybe eventually the entry deserves to be reorganized, but for the moment I have to quit and run.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 16th 2012

    added to category of sheaves the characterization

    (I had thought we had added these statements long ago, but now it seems to me that we haven't.)

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeOct 16th 2012

    I think that category of sheaves should treat not only the category of set valued sheaves but also very important categories of abelian sheaves (sheaves of 𝒪\mathcal{O}-modules) and the categories of quasicoherent sheaves. All of them in standard literature (and some others) are referred to as categories of sheaves.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeOct 16th 2012

    I can write these other cases, but the way the introduction of the page says is that it (seemingly?) does not allow!

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 16th 2012

    Possibly it makes sense to rename the page “category of set-valued sheaves”. A page “category of sheaves” could discuss a more general concept (and possibly splinter off to other pages, as for example the important case of sheaves of modules, etc.).

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 16th 2012
    • (edited Oct 16th 2012)

    We already have

    Maybe for the moment you could just add more cases as you desire without renaming too much. Currently many entries regard an unadorned “sheaf” by default as a “sheaf of sets”. So renaming will be a major undertaking to be done with care. And I am not sure if it is really necessary.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012

    Todd meant just to change the title of the very page to extend its scope. Urs, do you suggest that categories of sheaves of modules should be treated as separate entry (what I find somewhat cumbersome, as I think many features can be treated jointly for several different versions of categories of sheaves) or you allow to extend the entry categories of sheaves ? Notice that those categories have a number of related properties, e.g. the categories of quasicoherents sheaves of 𝒪\mathcal{O}-modules and topoi are both vectoids in the sense of Durov.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 17th 2012

    To create what Zoran wants, how about a page “sheaf valued in a category” or the like? It’s true that sheaf and category of sheaves concentrate on the SetSet-valued case.

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    Again I would prefer that a person who is outside of nnLab does not misses the topic he wants. Hence, I would like to add the content to the current page and not creating separate page for what is often a conventional usage for category of sheaves, as people who are used, say to abelian case, also use unadorned version as default. Who would guess to type “category of sheaves valued in a category” (yes, some of us, still most often look for a page by typing its supposed name and only if this fails go to long search). Also when accessing from other pages, one is not likely to refer with so specific long name. In fact the perspective of the page as it is now is on Gorthendieck topology case, hence the specialized title for such things if ever split would be better topos of sheaves or sheaf topos. For a topos of sheaves one understands at once “sheaves of sets” without a doubt. Category of sheaves would be more general than topos of sheaves in this sense. I’d also like to discuss categories of sheaves in noncommutative geometry which often fall beyond the Grothendieck topology case.

    I agree that the entry sheaf should also have a section and some remarks concerning the case of sheaves not of sets and will add this. I agree with Urs, that we do not need to specify “of sets” in most usages in nnLab but we should include/cover those cases at least in basic entries on sheaves. Hence no need of massive renamings, but need of occasional extra definitions. It is especially important at sheafification as it does not need to exist for presheaves in some other categories, and the cases when it does are important.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012

    or you allow to extend the entry categories of sheaves ?

    Not only do I allow it. I am hoping somebody will do it. And I’d hope both of this is clear.

    • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 17th 2012

    Now that Urs has explicitly given the green light, one could start by simply rewording the Idea section. Currently there’s

    This entry is about the properties and the characterization of the category Sh(S) of (set-valued) sheaves on a (small) site S, which is a Grothendieck topos. Among other things it gives a definition and a characterization of the notion of sheaf itself, but for more details sheaves themselves see there.

    One possibility is to rewrite this to

    This entry is mainly about the properties and the characterization of the category Sh(S) of (set-valued) sheaves on a (small) site S, which is a Grothendieck topos. Among other things it gives a definition and a characterization of the notion of sheaf itself, but for more details sheaves themselves see there.

    Sheaves valued in categories different from SetSet commonly arise and are also important; see section X for this.

    Obviously this is something off the top of my head and not terribly creative; it can be rewritten however Zoran wants to suit his purpose. E.g., if “mainly” is disliked, it could be “firstly” or something.

    • CommentRowNumber19.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012

    OK, I will work on this soon!!

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    Todd, Zoran,

    are you both being serious, or is this some kind of joke which I don’t get, when you write

    or you allow to extend the entry categories of sheaves ?

    and

    Now that Urs has explicitly given the green light

    ?

    What’s going on here? Do you think you need special permission from me in order to add content to the nnLab?

    On the contrary. I keep thinking to myself: “Why do Todd and Zoran write so much here in the nnForum discussion about trivial plans instead of just going ahead and creating content on the nnLab?”

    • CommentRowNumber21.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    Urs, I know you in general want us to add/change/improve and iterate this often. I do not doubt or ridicule that. But my plan of improvement here was specifically a bit orthogonal to the plan of the entry as it is now. Regarding that the entry is quite elaborate, hence you might be happy with accomplished as it is (it is not a stub!) I really care that we are here in agreement about major extension in scope.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    But you are not going to remove the discusssion of sheaves of sets, are you? So just add any further discussion either to the entry itself or to some dedicated separate entry linked to from there.

    By the way, I think that some years back when I wrote the first collection of entries on sheaf theory I had an entry on sheaves with values in other categories, following the section in Categories and Sheaves. But I can’t find it right now. Maybe I was dreaming.

    • CommentRowNumber23.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012

    No, in addition to new material/section, I plan to make only cosmetic changes to existing part, e.g. in few most basic definitions add “of sets” while leaving unadorned in more elaborate continuations of details and properties.

    • CommentRowNumber24.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 17th 2012

    @Urs #20: I was being serious, and I didn’t mean to give any offense whatsoever. I think we’re just being on the safe side, making sure no one is interfering with someone else’s plans. I am sorry if I said something to irritate you; I didn’t mean to.

    On the other hand, I don’t much like being told I write a great deal about trivial plans. I didn’t think that was true.

    • CommentRowNumber25.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012

    Urs 22: the entry sheafification has a section on sheaves with values in other categories, including discussion of IPC-property.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012

    Okay, good.

    Maybe for clarification: last time that I had complained about an edit the complaint was about changing something that I had written without announcing it here. But I don’t think that adding information to the nnLab can interfere with anyone’s plans. Adding information is the default purpose of the nnLab. The worst that can happen is that we later decide to reorganize the added information.

    the entry sheafification has a section on sheaves with values in other categories, including discussion of IPC-property.

    Ah, right, that’s what I remembered! Should all be linked more properly. But I can’t look into this right now.

    • CommentRowNumber27.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 17th 2012

    Okay, good.

    I’ll guess that part was addressed to me.

    Regarding #18, the way that should have been read is that I was talking mainly to Zoran and trying to offer a helpful suggestion about how to proceed in the editing.

    Back to business…

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