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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?
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.
Why does there appear to be an asymmetry in the treatment of each side of Isbell duality? I mean
Hmm, do copresheaves form a ’cotopos’? I remember asking about them a couple of years ago.
Much of this asymmetry is rooted in the fact that $Set$ 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.
We don’t have anything on ’copresentable’. Is local copresentability just about generating objects by limits?
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.
Hmm, do copresheaves form a ’cotopos’? I remember asking about them a couple of years ago.
$Set^C$, copresheaves on $C$, 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
$[Sh(C), Set]_{coc} \simeq CoSh(C) \,$so (edit!) when the contravariant Yoneda embedding $y$ lands in $[Sh(C), Set]_{coc}$, the image of the composite
$Sh(C)^{op} \xrightarrow{y} [Sh(C), Set]_{coc} \simeq CoSh(C) \,$will be a full subcategory of $CoSh(C)$ which is a “cotopos”, equivalent to $Sh(C)^{op}$. In general, you could restrict to those objects $S \in Sh(C)^{op}$ for which $y(S)$ is cocontinuous, and see what structure you get.
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?)
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
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.
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
@Colin #11
I wonder if I could get a hold of that? Mortensen is emeritus at Adelaide.
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 $C$ (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.)
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”.)
I tried it on Chrome and Safari and the same thing happens there!!!!
Hmm
$Presheaves$doesn’t seem to happen here.
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 )
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.
Looks good to me now, thanks!
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