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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeFeb 4th 2011

There is a major book available online in Spanish about the philosophy of contemporary mathematics. Quite a few contemporary mathematicians are mentioned in the book. The author is from Bogotá.

http://files.acervopeirceano.webnode.es/200000065-18c1b19bb9/Zalamea-Fil-Sint-Mat-Cont.pdf

MR2599170 Zalamea, Fernando Filosofía sintética de las matemáticas contemporáneas. (Spanish) [Synthetic philosophy of contemporary mathematics] Obra Selecta. Editorial Universidad Nacional de Colombia, Bogotá, 2009. 231 pp. ISBN: 978-958-719-206-3

• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeFeb 5th 2011

Should we list it on philosophy? (That page could really use a bibliography! David?)

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 5th 2011

Zalamea talks about my book a bit, so worth including ;)

Yes, I ought to make up a bibliography. I doubt there will be a settled way to consider n-categories for many a year, so a list of philosophical treatments might be the best we can do.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeFeb 8th 2011

New sections in philosophy, scope and most basic references. David, I asked via an email, which may have bounced, it came up in the discussion on another blog. If zou have some comment. What is the real importance of the Franch philosopher Alain Badiou in your opinion, and did he contribute any real results to math, as contrasted to just interpretations of math ?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 8th 2011
• (edited Feb 8th 2011)

When it comes to philosophy, I realize that I think of what it is about very differently than most of the professional practitioners.

At the very beginning when I heard of “philosophy of math” and “using $n$-category theory” in it I thought this is or should be (at least in parts) about using the striking connections that category theory makes between formal mathematics and ontological ideas. It can to some extent serve to formalize ontological ideas, it seems to me, providing a “math of philosophy” to some extent. Well, I am a naive amateur when it comes to philosophy. But at least it seems to me that I am not entirely alone with this idea. I may be wrong and may be all misunderstanding everyhting, but it seems to me that Lawvere has been pursuing this kind of “math of philosophy”-approach over his whole career.

It all starts with at the birth of category theory already with the question: what does it mean for a concept to be “natural”? Now, as we have seen here in earlier discussions, it may be debateable whether the formalization of natural transformation really well captures the ontological term “naturality”, but at least it is striking that category theory seems to allow to look for candidate formalizations at all. Whether one really finds them is maybe a different question.

Lawvere has tried many more such formalizations. He has tried formalizations of “being” and of “becoming”. Of “motion”. Of “quality” and of “quantity”. Things like that. I suppose all the suggestions that he has made for what the formalizaitons of these should be are not set in stone in any way, first of all he himself seems to have been trying to fine-tune the notions over the years. Which is only natural.

But whether or not there have been widely accepted “results” in such “categorical semantics of ontology” or whatever it should be called, isn’t it most striking and pleasant that it can be considered at all? Isn’t there a strong subject “categorical philosophy” or the like to be investigated by (other) philosophers of math/science/the universe here?

I think I would enjoy seeing more along these lines on the $n$Lab’s so far pretty dormant third topic.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeFeb 8th 2011

It all starts with at the birth of category theory already with the question: what does it mean for a concept to be “natural”?

Isn’t even this still about metaphysics of mathematical concepts ?

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeFeb 8th 2011
• (edited Feb 8th 2011)

I decided, that with growing of the philosophy main page the query box, previously encapsulated as a regular section by Gavin Wraith may be more useful standing instead here on nForum (with a backlink at philosophy), as it is already in the form of a discussion. So here it is

The dangers of category theory

Gavin: This is a sensitive topic. I do not know if this is the right place for it, but I feel that $n$Lab would be lacking balance if it did not have some part that attempted to address the fact Category Theory has been responsible for some pretty wretched mathematics. To go into too much detail might be discourteous. I was delighted by the beauty of category theory when I first met it (in the library of the Courant Institute in 1962, when I was supposed to be writing a thesis in Theoretical Physics). However, I soon realized that in the mathematical community of the UK, at least, at that time, too great an indulgence in abstraction was viewed with suspicion. It is a suspicion that I share, to some extent. I remember coming across an article, some years later, in a Physics journal, written by a well-respected physicist, in which large chunks of a paper by Applegate on the Tierney Tower of adjoint functors were repeated verbatim; and evidently with the corrupt intention of combombulating readers. Category theory is easily misused, either for unnecessary flights into the Empyrean, or for more evil El-Naschiean purposes. But I guess that most nLabians are perfectly aware of all this. Since those early days category theory has become a lingua franca for almost every branch of mathematics. I too love abstraction; but I am just a little bit wary of it.

If you came across a journal called Category Theory and Prosopography would you smell a rat? The difficulty with cross-subject journals is that they make it too easy for authors to evade proper peer-review. Told that the mathematics in his paper is crap, the author can try to wriggle out by asserting that it is a masterpiece of prosopography. The shock and awe of the reputation of mathematics makes it less likely that prosopographers will condemn his paper from their side. It is just because Category Theory binds together widely disparate branches of mathematics that it lends itself to the snake-oil merchant.

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeFeb 8th 2011

More links and slight change in sectioning under foundations and logic.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 8th 2011

I’m feeling guilty about not having put up more about philosophy on nLab, but may have a chance to do something about this, since I’ve been asked to give some reflections on n-categories to a seminar in Paris next month. I’m certainly sympathetic to Urs point that Lawvere (and so Urs himself) is doing some variety of philosophy:

It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy. (Categories of Space and Quantity, 1992)

There’s an interesting question of what the philosophical training does for you in the context of the final sentence. Why should it help a philosopher develop Lawvere’s ideas? Why should it help a mathematician? Learning maths on the other hand is a different matter.

There’s a kind of ’doing philosophy’, especially on the leading edge of a wave of change, which is best done by a certain kind of mathematician-physicist. Poincare is a good example, and Einstein in his way too. And then there’s a tidying up kind of philosophy after the wave has gone. Perhaps Kant is a good example of that. We probably most need our Poincare type at the moment, as far as n-categories go, and then, in a way, they are doing philosophy merely by doing their thing, e.g., writing about space and quantity, and the duality between them. What does a philosopher add to that?

I’m going to have to knuckle down to these question over the next few weeks.

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeFeb 8th 2011

More changes under foundations and logic.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeFeb 8th 2011
• (edited Feb 8th 2011)

Thanks, David.

I have never tried to study Lawvere’s thoughts in this direction in any systematic way, but whenever I come across his – usually somewhat vague – remarks it strikes me that he is evidently after something that deserves to be pursued, and which few other people are actively pursuing.

Why should it help a philosopher develop Lawvere’s ideas? Why should it help a mathematician?

I cannot say anything about the former. But maybe I have an idea about the latter:

it is necessary for finding the right structures to concern oneself with.

It is good that you mention Einstein and Poincaré, they serve as good examples for what I mean at a foundational level a good bit higher up in the hierarchy of things than Lawvere has been aming at:

together, back at their time, they answered the question: what is the structure to study for describing space and time on large scales?

The answer was of course: differential geometry. Ever since mathematicians can pick up a book and “work in differential geometry”. But at some point in the past somebody must have figured out that among all books to pick up, this one is the correct one.

Right as we speak, in these years, we are witnessing the precisely analogous question being answered a level further down in the hierarchy of things: what is the structure that described geometry more generally. (Such that “differential geometry” is but one example.)

Long time ago it has become clear that the answer will involve sheaves. Then it has become clear that it will also involve co-presheaves. But how exactly? We can study all sorts of axioms as mathematicians, but which of them “correctly” encapsulates the notion geometry that we do want to study.

Lurie’s “Structured Spaces” is one attempt at an answer. Lawvere had been addressing this very issue for decades already. He amplified that the duality between presheaves and copresheaves should play a central role, and various other aspects.

Once the right answers here have been found, the axioms are set up and it is all math. Just as with Einstein: once he had found the framework now called “Einstein gravity” from that point on it was just all standard theoretical physics: take the axioms and see what happens.

But to find the right axioms, one has to step out of the formalism, necessarily. That is not math, that is not physics, by definition. That is probably then philosophy.

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeFeb 8th 2011
• (edited Feb 8th 2011)

Lurie’s “Structured Spaces” is one attempt at an answer.

I will be happy once a fully noncommutative case will be treated as well. Unfortunately quasicoherent sheaves on a noncommutative space do not seem to form something close enough to topoi, and the generalizations of Grothendieck topology also have quite distinct features than on commutative spaces. I have some ideas how to start with investigation for a common generalization, but very rudimentary.

Lawvere had been addressing this very issue for decades already. He amplified that the duality between presheaves and copresheaves should play a central role, and various other aspects.

Nikolai Durov told me recently that he has some motivation for “spaces” which are not determined by any kind of generalized (including higher categorical) algebra of functions, topos of sheaves etc. and that he has some developing proposal for such a geometry. I do not know what he actually thinks but I am sure it will be something very original.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeFeb 9th 2011
• (edited Feb 9th 2011)

I will be happy once a fully noncommutative case will be treated as well.

It seems to me that this will be nicely incorporated in $\left(\infty ,2\right)$-topos theory.

We have discussed this here and there before: it quite seems that the noncommutative geoemtry program at least in the style of Kontsevich’s school is well thought of as dealing with spaces formally dual to commutative $\left(\infty ,2\right)$-algebras, namely to certain monoidal $\left(\infty ,1\right)$-categories (“of quasicoherent sheaves on a would-be noncommutative spaces”).

This seems to me to be quite a compelling point of view.

Is there no chance that there is a reasonable Grothendieck topology on some small version of the opposite of the $\infty$-category of suitably well behaved symmetric monoidal $\infty$-categories? It would seem to me to be quite compelling that noncommutative geoemtry would naturally live in the $\infty$-topos over such a site.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 9th 2011

Re 11, to go back to the theme of prospective and retrospective forms of philosophy, it’s noticeable that the possibly more creative prospective forms reach out in a somewhat unsystematic way to other fields. For example Poincare, along with Helmholtz and others, was very interested in a variation of Kantianism, where instead of Euclidean geometry being built into our spatial intuition, we come into the world equipped with a range of possible geometries, experience determining which we end up with. So they explored possible geometries of constant curvature

that there is an a priori intuitive basis for geometry in general, upon which the different metric geometries can be constructed in pure mathematics. Once constructed, they can then be applied depending on empirical and theoretical need. The a priori basis for geometry has two elements for Poincaré. First, he postulated that we have an intuitive understanding of continuity, which – applied to the idea of space – provides an a priori foundation for all geometry, as well as for topology. Second, he proposed that we also have an a priori understanding of group theory. This additional group theoretic element applied to rigid body motion for example, leads to the set of geometries of constant curvature. here

Now we recognise the part of Poincare in the emergence of a new geometric understanding of the universe, but typically we don’t care much about his Kant-inspired geometric psychology. I think this is a common story, and I wouldn’t mind betting that people in 2100 will appreciate Lawvere’s role in the emergence of category theoretic understanding, but will not be interested in the Hegelian inspiration.

• CommentRowNumber15.
• CommentAuthorzskoda
• CommentTimeFeb 9th 2011
• (edited Feb 9th 2011)

We have discussed this here and there before: it quite seems that the noncommutative geoemtry program at least in the style of Kontsevich’s school

Yes, and I am repeatedly repeating: this is the derived case which is essentially commutative (no noncommutative Grothendieck topologies needed etc etc). Not a big deal from the point of view of noncommutativity at the level of abelian categories. The passage from the abelian category to its derived category (in any of its enhanced versions) is essentially loosing most of noncommutativity. The derived level does not really distinguish commutative from noncommutative.

I am working for 15 years on the full noncommutativity and see no touch of essential problems at abelian level.

Is there no chance that there is a reasonable Grothendieck topology on some small version of the opposite of the $\infty$-category of suitably well behaved symmetric monoidal $\infty$-categories?

The abelian categories of qcoh sheaves on noncommutative schemes are not monoidal (though there are such approaches for bimodules, but I think it should be interpreted differently). We do have candidates for noncommutative Grothendieck topologies, but they are not Gorthendieck topologies, they satisfy weaker axioms, and the categories of set-valued sheaves do not form Grothendieck topoi.

Most importantly, the pullback of a cover is not a cover in general.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeFeb 9th 2011
• (edited Feb 9th 2011)

this is the derived case which is essentially commutative (no noncommutative Grothendieck topologies needed etc etc)

You seem to say “essentially commutative” for “the theory behave as in the commutative case”. That sounds like an advantage to me, not a disadvatage! It sounds like for this setup the theory does work and manages to descibe noncommutative spaces on par with commutative ones.

The passage from the abelian category to its derived category (in any of its enhanced versions) is essentially loosing most of noncommutativity.

What’s an example of an application where one cares about infomation pesent in the abelian category and not its $\infty$-category?

The abelian categories of qcoh sheaves on noncommutative schemes are not monoidal (though there are such approaches for bimodules, but I think it should be interpreted differently). We do have candidates for noncommutative Grothendieck topologies, but they are not Gorthendieck topologies, they satisfy weaker axioms, and the categories of set-valued sheaves do not form Grothendieck topoi.

Most importantly, the pullback of a cover is not a cover in general.

Somebody might take all this as indication that something is wrong about the setup then; that it all needs to be fixed, apparently by passing to the corresponding $\infty$-categoies of qcoh sheaves.

So there must be strong motivation fom examples that suggests that despite all the trouble with its theory, this setup that you have in mind is still a good one to consider. What are these examples?

• CommentRowNumber17.
• CommentAuthorzskoda
• CommentTimeFeb 9th 2011
• (edited Feb 9th 2011)

It sounds like for this setup the theory does work and manages to descibe noncommutative spaces on par with commutative ones.

Urs, there is a forgetful functor from abelian to derived setup: even a commutative scheme can be reconstructed up to isomorphism from the abelian category of quasicoherent sheaves but not from the derived category (nor any stable enhancement), in general. Further you are from a commutative algebra, the deriving looses more information.

Somebody might take all this as indication that something is wrong about the setup then; that it all needs to be fixed, apparently by passing to the corresponding ∞-categoies of qcoh sheaves.

There is a well-defined subject of noncommutative algebraic geometry. Forgetting its essential features is not *fixing\$ in my opinion. For many purposes in physics and so on, real hardcore noncommutative geometry is not relevant and the derived picture suffices, but there are many, even easy questions, for example the definition and study of nonabelian Čech cohomology on noncommutative spaces which has features unknown in commutative case and which is essential if one wants to study Galois theory for noncommutative rings in its full subtlety.

So there must be strong motivation fom examples that suggests that despite all the trouble with its theory, this setup that you have in mind is still a good one to consider. What are these examples?

It is difficult to find a noncommutative algebra which is not an example of these phenomena. There is a useful category of quasicoherent modules and the noncommutative localizations serve as open sets. However they do not form a Grothendieck topology. There are very useful examples of torsors and so on defined with respect to such topologies. See for example my article on coherent states, pdf.

apparently by passing to the corresponding ∞-categoies of qcoh sheaves

The difference between some Galois extensions may get annihilated by passing to the derived picture. One needs to stay at abelian level. On the other hand, I agree that the derived level for sheaves of sets is useful, in order to include a larger class of localizations. This does not solve the problem with pullbacks but it does provide a larger class of examples and larger flexibility in finding needed localizations, especially when treating very large algebras (those close to the free ones).

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeFeb 9th 2011

But what’s a motivating example? Some question where the $\infty$-category version somehow does not help?

• CommentRowNumber19.
• CommentAuthorzskoda
• CommentTimeFeb 9th 2011
• (edited Feb 9th 2011)

Classify “vector bundles” for example with structure Hopf algebra or if you like Hopf-algebrtaic torsors. E.g. the bundles with quantum groups as structure groups. There are geometric methods in representation theory, involving bundles and sheaves and their nonabelian cohomologies over noncommutative spaces. Try for example to represent the derived functors for induction functor for representations of quantum groups as Čech cohomology of some sheaves on quantum flag varieties. There is a problem with refinements here, very noncommutative phenomenon (I mean limiting constructions in Čech cohomology).

• CommentRowNumber20.
• CommentAuthorzskoda
• CommentTimeFeb 9th 2011
• (edited Feb 9th 2011)

Topology is used in algebraic examples to pass between local and global. In defining and calculating with quantum group coherent states I used in loc. cit. together local formulas and global theory (characters, Haar measure etc.). Locality is in the sense of noncommutative localization. Now I found the method very useful. In fact this was the only method I could find after few yars of thinking which lead to a unique (up to overall normalization) invariant coherent state measure which disagrees with the eralier papers which used ad hoc deformations without theory (hence loose invariance both of the measure and of the integrand). I want to continue using such covers in systematic way to study quasicoherent sheaves in noncommutative setup. Loosing this for some ideology of infinity categories is a sacrifice I can not afford, as a practioner from quantum group theory.

Of course, we have here two different kinds of theory, I mean the big site and the small site; I was talking about the small “site” here, but similar problems are in the big site picture, which I understand much less.

• CommentRowNumber21.
• CommentAuthorzskoda
• CommentTimeFeb 10th 2011
• Ralf Krömer, Tool and object: A history and philosophy of category theory, Birkhäuser 2007

added to philosophy books. Somebody has some comments on its value ?

• CommentRowNumber22.
• CommentAuthorzskoda
• CommentTimeFeb 10th 2011
• (edited Feb 10th 2011)

To add to 20 (discussion with Urs), there are spectra of noncommutative spaces which are real topological spaces with structure stacks; however such constructions, if faithful (determining the space) are not functorial for all types of morphisms (usually they are functorial with respect to some subclass of morphism). This points to the fact that the topological/topos/locale plus sheaf/stack picture needs some refining to absorb noncommutative spaces in nonderived version. Of course, there is a basic belief in the noncommutative community that once, in 23rd century, we will have some sort of genuine spaces behind noncommutative geometry of any kind, with all the flexibility like in the commutative case…but it needs a nontrivial work in nonderived case. Which is interesting, I mean the generalizations of cohomology and sheaf theory to things which are not quite Grothendieck topologies is very interesting problematics.

• CommentRowNumber23.
• CommentAuthorDavidRoberts
• CommentTimeFeb 11th 2011

@Zoran,

I have vaguely read what you have written about these not-quite-Grothendieck-(co)topologies, but what is the real distinguishing feature that means they are not topologies. I know you said that a pullback of a cover (or cocover, depending on your perspective) is not a cover, but I’m just checking that what you have is not a coverage.

Are there links to quantum categories of Street et al? Are there some really basic examples of these not-topologies (like for the category of non-commutative rings)? Do the properties of the (co)covers make sense in any monoidal category?

I’m sorry for asking such basic questions, but I’m intrigued. One thing which I might look into at some point is redoing my work on anafunctors but with quantum categories instead of plain internal categories (which are quantum categories for the case of a cartesian monoidal category). Maybe it won’t work at all, maybe it will. But I’m sure there’s something interesting to get out of it. In particular, what sort of localisations of (2-?)categories of Hopf algebroids will this give rise to?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeFeb 11th 2011
• (edited Feb 11th 2011)

Classify “vector bundles” for example with structure Hopf algebra or if you like Hopf-algebrtaic torsors.

Okay, thanks. I looked for some references and have collected what i saw now at quantum vector bundle. But I guess I am missing important references (only had a few minutes).

In these articles that I found it seems that only quantum vector bundles over the spec of a single nc algebra are considered.

Could you point me to some result that would illustrate the phenomenon that you mentioned, that there is a loss of important information when passing to the oo-categories of quasicoherent sheaves on th non commutatiev base space on which one is interested in quantum vector bundles?

• CommentRowNumber25.
• CommentAuthorzskoda
• CommentTimeFeb 11th 2011
• (edited Feb 11th 2011)

But I guess I am missing important references (only had a few minutes).

More strongly: the references you put are more of a distraction than use for the central questions of noncommutative topology we discuss above (I apperaciate those authors for other aspects though). Zhang is working in C-star algebraic setup, not algebraic, so not good for the study of quasicoherence. One of his papers has errors in the treatment of involutions as far as I recall. Coquereaux etc. is not very relevant – it just defines, in the affine case, examples for the global sections of what would be an associated bundle, not the associated bundle per se. This is avoiding the main problem, as I did in some early references.

I have written ages ago more relevanty entry noncommutative principal bundle.

In these articles that I found it seems that only quantum vector bundles over the spec of a single nc algebra are considered.

I was telling you above and many times before about my own work which in 1997 went beyond that, e.g. the work on quantum group coherent states. Some useful background is also at noncommutative scheme and gluing categories from localizations (zoranskoda).

• CommentRowNumber26.
• CommentAuthorzskoda
• CommentTimeFeb 11th 2011
• (edited Feb 11th 2011)

but what is the real distinguishing feature that means they are not topologies

The category of set valued sheaves is not left exact reflective subcategory of the category of presheaves. Left exact must be dropped in most examples. On the other hand, for abelian sheaves one has the problem with pullbacks of covers. One relevant paper is Rosenberg’s Noncommutative schemes paper (cf. entry noncommutative scheme). Look also for works of van Oystaeyen and his school including his book on nc geometry for associative algebras, and my article on equivariant aspects and my earlier survey on nc localization in nc geometry. Quantum groupoids aka Hopf algebroids are just more general group/like objects which can take place of a structure group. I think Maszczyk has many new examples in that direction. However nothing essentially new in the sense that already for usual Hopf algebras one has all the difficulties with local trivializations.

• CommentRowNumber27.
• CommentAuthorzskoda
• CommentTimeFeb 11th 2011
• (edited Feb 11th 2011)

Could you point me to some result that would illustrate the phenomenon that you mentioned, that there is a loss of important information when passing to the oo-categories of quasicoherent sheaves on th non commutatiev base space on which one is interested in quantum vector bundles?

By Gabriel-Rosenberg a commutative scheme (with some very mild restriction) can be reconstructed from the abelian category of qcoh sheaves. This is not true from the derived category except in extremely special cases treated e.g. by Bondal-Orlov theorem for projective varieties when the canonical or anticanonical line bundle is ample. There is no essential difference for the reconstruction if one takes stable $\left(\infty ,1\right)$-version of derived category – Orlov and Lunts have shown a couple of years ago for smooth quasiprojective varieties that such an enhancement is essentially unique, hence there is no difference in information weather I take old-fashioned derived category or stable quasicategory or A-infty. The derived reconstruction is somewhat better if one keeps the tensor structure, what one does not have in noncommutative case in general.

• CommentRowNumber28.
• CommentAuthorzskoda
• CommentTimeFeb 11th 2011
• (edited Feb 11th 2011)

The continuation of noncommutative bundle discussion could continue better here, and this track left for philosophy books…

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeFeb 17th 2011

Hi Zoran,

By Gabriel-Rosenberg a commutative scheme (with some very mild restriction) can be reconstructed from the abelian category of qcoh sheaves. This is not true from the derived category except in extremely special cases

Right, that’s what you said. what I would like to know is concrete examples where one does care about the information that is not seen by the derived or $\infty$-category of quasicohrent sheaves. What kind of information is lost and in which examples is it essential to keep that information?

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeFeb 17th 2011

I need to be looking a bit into the noncommutative Gelfand-Naimark theorem in the topos of copresheaves over the poset of commutative subalgebras of a noncommutative $C*$-algebra. I still need to make up my mind on how to best think of this sizuation and the duality result and the grand hopes that some people have about this and how it all fits into all the rest. Not sure yet.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeFeb 17th 2011
• (edited Feb 17th 2011)

It’s maybe interesting for our conversation that in that context, of the internal GN theorem, too, passing from a non-commutative algebra to its poset of commutative subalgebras loses some of the information: under some conditions only the Jordan algebra of the noncommutative algebra is remembered. (As discussed here)

It would be interesting to better understand what that means. It seems that for the non-commutativity of an algebra of observables in quantum theory, in fact the Jordan algebra structure is all that really matters. Maybe. Which would mean that the topos-theoretic description remembers precisely the right aspect of the non-commutativity.

Just an observation. I need to better understand all this.

• CommentRowNumber32.
• CommentAuthorzskoda
• CommentTimeFeb 17th 2011
• (edited Feb 17th 2011)

What kind of information is lost and in which examples is it essential to keep that information?

As this is about the reconstruction of commutative schemes, then you are asking me about the commutative geometry. I do not have much knowledge on commutative algebraic geometry, and what I got from others is that this in the most classical case of finite type/varieties some finite information which is not very essential, even in the sense of classical Italian school. In noncommutative situations one looses much more. For example, the noncommutative projective space in the sense of Kontsevich and Rosenberg, which is really huge and locally looks almost like a free noncommutative algebra, has the same derived category of coherent sheaves as the commutative projective space. The first has as subvarieties various guys like quantum projective spaces, while the latter does not have them. This is really cruel.

By the way, Urs, I have in 28 given a link where should the discussion on noncommutativity continue. My feeling is that nc Gelfand-Neimark also looses some info even in operator algebraic framework which is by its nature more commutative anyway.

Which would mean that the topos-theoretic description remembers precisely the right aspect of the non-commutativity.

Again you are in operator algebras, what is really not that noncommutative. I agree that for operator algebras the derived picture looses very little (in part because of the existence of involution and of boundedness), unlike in algebraic situation. I am really interested in true noncommutative algebraic spaces, and I do not find any satisfactory treatment from the derived camp yet. I mean even for the main and simplest examples of my interest.