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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!
We were talking about cotoposes above…
More broadly, we could regard categories of space as distributive or extensive or lextensive (that is, let us momentarily take a step back from toposes). What is “codistributive”? Of course, the answer is “opposite category of distributive”, but how do we think of such categories? I would hesitantly contend that “codistributive” is “algebra-like”. For instance, coproducts distribute over products in $R \text{-alg}$. Also, $R \text{-alg}$ is coextensive (opposite category of an extensive category) - precisely the reverse from spacial categories. This “algebra-like” codistributivity is of course quite different than “linear” - it reminds more of rings/algebras than modules.
This observation is more or less expected if “algebra” categories stand in a sort of duality with “spacial” categories. But why do categories of algebras only seldom have “cosubobject classifiers”. I don’t know, nor do I know what it means to be “co-cartesian closed” (this seems like it could actually be more subtle than “opposite category of a cartesian closed category”).
Nevertheless, to continue, we may further observe that Isbell duals tend to be between “distributive” and “codistributive”. They are also often between linear (coproducts canonically isomorphic to products) and colinear, but the opposite of a linear category is of course the same as a linear category. For an example, take $R$-mod for a commutative ring $R$ and $\text{Hom}(-, M)$.
Contravariant Isbell duality is $[C, V] \leftrightarrow [C^{op}, V]$ when $C$ is enriched over $V$, where the brackets denote enriched functors. There is no mention of distributive, extensive, codistributive, topos, etc., which are yet to be supplied by specific context. Accordingly, we might notice that
1) If $V$ is distributive, then $[C, V]$ is distributive.
2) If $V$ is codistributive, then $[C, V]$ is codistributive.
3) If $V$ is linear, then $[C, V]$ is linear.
So it is the nature of the target category that tells the nature of the (co)presheaves. Of course, taking $C = D^{op}$ for another category $D$ will make no difference.
Now I am curious about putting conditions on $[C, V]$ such as limit preservation. Perhaps requiring limit preservation inverts type and colimit preservation preserves type? By type I mean, “spacial” (products distribute over coproducts), “linear” (products and coproducts canonically isomorphic), or “algebraic” (coproducts distribute over products).
Also, you guys will probably know the answer to this: if (the contravariant) Isbell duality pertains to $\text{Hom}(-, X)$ with $X$ playing two roles, then what is an analogously general duality between $\text{Hom}(X, -)$ and $X \otimes -$ with $X$ playing two roles?
P.S. I can’t seem to get Markdown+Itex to work for me. Has anyone else encountered this on Safari?
Also, cohesion seems to exist for each type- there is no reason to prefer the spacial type as far as I can see. For instance, there is a cohesion between modules and topological modules. The functors are “discrete module”, “underlying module”, “indiscrete module”, and “$M$ maps to $M/N$ where $N$ is the smallest submodule containing the connected component of $0$”. Or for another example, take commutative rings and the full subcategory of functors $[I, \text{Ring}]$ from the interval category to ring consisting of surjective maps of nilpotent ideals. I think that the broader view here is that functors in cohesion preserve type, which follows for the three examples of type I gave.
David- from some of your posts you found it seemed like you share my fascination for making mathematics symmetrical like this, so let me know if you want to work on it with me.
I was wondering about some kind of cohesive modality-algebra link over here based on an observation mentioned in #21, #44 there of a 7-term chain of adjoints.
I’m an interested observer of such general patterns. There were discussions on coalgebra over at the n-Café once, e.g., here. Not sure I have anything much to add at present, but I’m happy to see you think out such matters.
Added:
The codensity monad of the Yoneda embedding is isomorphic to the monad induced by the Isbell adjunction, $Spec \mathcal{O}$ (Di Liberti 19, Thrm 2.7).
Is there anything to stop the Yoneda embedding for bicategories similarly generating an Isbell adjunction? The results on the page are written for enriched categories, so I guess no problem for strict 2-categories.
Kan extensions for $Cat$ are as in Enriched categories as a free cocompletion, sec. 10?
I expect all of this would go through without much change for higher categories of all sorts.
Thanks. So perhaps Gabriel-Ulmer duality and the ultracategory one I’m talking about here can all be construed in Isbellian terms.
Just to check, after Proof B where it says
The pattern of this proof has the advantage that it goes through in great generality also on higher category theory without reference to a higher notion of enriched category theory,
can I add
for instance, employing the 2-Yoneda lemma, there is an Isbell duality between presheaves and copresheaves of categories on a small 2-category?
Sure, that’s exactly the way I think of it!
This is also true for enriched categories, right?
I am having trouble understanding some of the notation here. We have a $V$-enriched category $C$ but then the expression $V(-,c)$ is used in many places; I assume this is meant to refer to the V-object $C(-,c)$? Or is this some common notational convention I was unaware of? If people are ok with $C(-,c)$ I’d be happy to change it, just wanted to make sure I’m not stepping on anyone’s toes.
That might have been me, back then. Please fix as seems appropriate.
I have added (here) the doi-link doi.org/10.1090/noti2602, since that should eventually be the more stable one. On the other hand, it doesn’t work at the moment, but I guess they will fix this eventually.
added links for these references:
{#Isbell66} John Isbell, Structure of categories, Bulletin of the American Mathematical Society 72 (1966), 619-655. [project euclid, ams:1966-72-04/S0002-9904-1966-11541-0]
{#Isbell67} John Isbell, Normal completions of categories, Reports of the Midwest Category Seminar, 47, Springer (1967) 110-155 [doi:10.1007/BFb0074302]
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