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    • CommentRowNumber1.
    • CommentAuthorspitters
    • CommentTimeJan 20th 2014

    http://ncatlab.org/nlab/show/Isbell+duality

    Suggests that Stone, Gelfand, … duality are special cases of the adjunction between CoPresheaves and Presheaves. A similar question is raised here. http://mathoverflow.net/questions/84641/theme-of-isbell-duality

    However, this paper http://www.emis.ams.org/journals/TAC/volumes/20/15/20-15.pdf

    seems to use another definition. Could someone please clarify?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 21st 2014
    • (edited Jan 21st 2014)

    Isbell duality is a conceptual template for the other space/algebra-dualities, but the latter are not in general direct special cases without further bells and whistles. This is already so for the simple example of function T-algebras there, where one restricts to product preserving co-presheaves, and then more so for function T-algebras on sheaves for which on the one side one restricts to product preserving co-presheaves and on the other to presheaves that are sheaves.

    To clarify I have now renamed “Examples” to “Examples and similar dualites” and added a lead-in sentence along the above lines.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 21st 2014

    Why does there appear to be an asymmetry in the treatment of each side of Isbell duality? I mean

    • The pages for presheaf and sheaf are much more extensive than for copresheaf and cosheaf.
    • There is a nice informative page motivation for sheaves, cohomology and higher stacks, but not for cosheaves.
    • space and quantity starts off with “a category C whose objects are thought of as spaces”. Why not tell a story beginning with a category of quantities?

    Hmm, do copresheaves form a ’cotopos’? I remember asking about them a couple of years ago.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2014

    Much of this asymmetry is rooted in the fact that SetSet is a locally presentable category, but not co-presentable. One may wonder why that is important, if it is. In type theory this is related to emphasis of inductive types over coinductive types. If one were to get to the bottom of the asymmetries that you mention, I suppose one would have to make up ones mind about this “root” of it all.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 21st 2014

    We don’t have anything on ’copresentable’. Is local copresentability just about generating objects by limits?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2014

    Yes, I just mean representability of the opposite category.

    I should say that I don’t have any sophisticated thoughts on co-representability, others here may have much more to say. The main point I wanted to highlight is that the choice of sets as the basis of it all involves an asymmetry in which role limits and colimits play, and it is this asymmetry which percolates through to all those that you mentioned.

    The category theorists, logicians and type theorists who are around here (or were?) should be able to give a much better comment on this, though.

    • CommentRowNumber7.
    • CommentAuthorColin Zwanziger
    • CommentTimeMay 24th 2017
    • (edited May 25th 2017)

    Hmm, do copresheaves form a ’cotopos’? I remember asking about them a couple of years ago.

    Set CSet^C, copresheaves on CC, is a topos, not a “cotopos” (opposite category of a topos).

    However, we might hope for a relation between cosheaves and cotoposes, analogously to the relation between closed sets and coHeyting algebras.

    Cosheaf states that

    [Sh(C),Set] cocCoSh(C) [Sh(C), Set]_{coc} \simeq CoSh(C) \,

    so (edit!) when the contravariant Yoneda embedding yy lands in [Sh(C),Set] coc[Sh(C), Set]_{coc}, the image of the composite

    Sh(C) opy[Sh(C),Set] cocCoSh(C) Sh(C)^{op} \xrightarrow{y} [Sh(C), Set]_{coc} \simeq CoSh(C) \,

    will be a full subcategory of CoSh(C)CoSh(C) which is a “cotopos”, equivalent to Sh(C) opSh(C)^{op}. In general, you could restrict to those objects SSh(C) opS \in Sh(C)^{op} for which y(S)y(S) is cocontinuous, and see what structure you get.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 25th 2017
    • (edited May 25th 2017)

    Thanks. You’d get a badge on MO for reviving an old question.

    Does cotopos merit a page? We do have a mention of the ’dual of a topos’ being a protomodular category.

    (By the way, why does ’coc’ stand for colimit preserving?)

  1. Does cotopos merit a page?

    I’m not sure “cotopos” is in use, but the terminology seems inevitable. Can’t keep calling it “dual of a topos” forever. The cosheaf case looks like a setting where a cotopos has the to opportunity to interact with some non-dual gadgets.

    (By the way, why does ’coc’ stand for colimit preserving?)

    coc \equiv cocontinuous \equiv colimit-preserving

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 25th 2017

    I’ll explain that at cosheaf.

    I mentioned two appearances of ’cotopos’ back here. A further one comes in a chapter on paraconsistent logic.

  2. Your link also mentions the notion appears in

    W.James and C.Mortensen, Categories, sheaves and paraconsistent logic

    which elsewhere is mentioned as dated 1997.

    • CommentRowNumber12.
    • CommentAuthorMatt Earnshaw
    • CommentTimeMay 25th 2017

    perhaps worth noting that “complement topos” is used in various places to describe constructions whose internal logic is “dual intuitionistic” (paraconsistent), and though they have some dual aspects of structure (eg. a coheyting “dual classifier”) they are not formally dual to toposes in the usual sense of reversing all arrows in the definition

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 25th 2017
    • (edited May 25th 2017)

    @Colin #11

    I wonder if I could get a hold of that? Mortensen is emeritus at Adelaide.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2018

    there was no pointer back to space and quantity from here. I have added the following paragraph to the Idea-section:

    Under the interpretation of presheaves as generalized spaces and copresheaves as generalized quantities modeled on CC (Lawvere 86, see at space and quantity), Isbell duality is the archetype of the duality between geometry and algebra that permeates mathematics (such as Gelfand duality or the embedding of smooth manifolds into formal duals of R-algebras.)

    diff, v24, current

    • CommentRowNumber15.
    • CommentAuthorAli Caglayan
    • CommentTimeJan 6th 2019
    I am on firefox and I see the "PreSheaves" in the formula as "Prʃeaves". The renderer seems to be picking up "eSh" and literally making it an esh.
    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJan 6th 2019

    I see that too! (It’s the first displayed formula on the Isbell duality page in the Idea section.) What a weird bug. (In the source it’s actually “esh” not “eSh”.)

    • CommentRowNumber17.
    • CommentAuthorTim_Porter
    • CommentTimeJan 6th 2019

    I tried it on Chrome and Safari and the same thing happens there!!!!

    • CommentRowNumber18.
    • CommentAuthorMarc
    • CommentTimeJan 7th 2019
    Strange. I tried it with Lynx, and it seems to be in the source. Perhaps somebody has used an editor that expanded the 'esh' keystrokes into 'ʃ' via some makro and did not notice this during the edit.
    • CommentRowNumber19.
    • CommentAuthorAli Caglayan
    • CommentTimeJan 7th 2019
    • (edited Jan 7th 2019)

    Hmm

    PresheavesPresheaves

    doesn’t seem to happen here.

    • CommentRowNumber20.
    • CommentAuthorRodMcGuire
    • CommentTimeJan 7th 2019

    It appears to be the fault of the new renderer. The diff view, which uses the old renderer, works correctly.

    https://ncatlab.org/nlab/show/diff/Isbell+duality

    ( arrgh I can’t use https://ncatlab.org/nlab/show/diff/Isbell+duality/25 to link the current diff in case somebody updates the file )

  3. I have fixed it now, hopefully.

    The symbol \esh was one of the LaTeX symbols I added support for a while ago, along with a few others. The way it was done was not quite robust enough, I have tried to tighten it now. They are not present in the original Itex2MML, and are currently added by post-processing.

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeJan 7th 2019

    Looks good to me now, thanks!

    • CommentRowNumber23.
    • CommentAuthorFosco
    • CommentTimeJan 8th 2019
    I'll just leave this here :-)

    https://mathoverflow.net/questions/313487/a-serendipitous-connection-between-isbell-duality-and-yoneda-structures
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