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Added (not very elegantly) the relation to Meyer-Vietoris sequence at Dold-Thom theorem.
tried to add some basic formatting. And some more links.
It would be good to mention that $Sym^\infty$ is a linear functor, but as I mention this would be ambiguous.
Regarded the picture with (infinity,1)-tangent category of Lurie, a term of the sort derived linear would look better. One should also think of enrichment over spectra as generalizing the enrichement over vector spaces. Right ?
Does this have something to do with just as spectra form the (infinity, 1)-tangent category for spaces at the one point space, $Vect_F$ is the tangent category of $CRing$ at the field $F$?
I guess so, and one would like to always approximate by such a situation, like the derivative of a mapping between Banach spaces or manifolds approximates a general map by a linear map.
One should also think of enrichment over spectra as generalizing the enrichement over vector spaces. Right ?
Strictly speaking, enrichment in spectra is the analog of enrichment inm abelian groups. A spectrum is exactly an $\infty$-abelian group in a way.
And $Ab$ is the tangent category of $CRing$ at $\mathbb{Z}$.
And $Ab$ is the tangent category of $CRing$ at $\mathbb{Z}$.
Wow. That sounds neat. I’m not sure if this question makes sense, but is $CRing$ somehow “smooth” with infinitesimal morphisms? I’m just thinking of the statement “a Lie algebra is the tangent space at the identity” and wondering if there is any relation.
David writes:
And Ab is the tangent category of $CRing$ at $\mathbb{Z}$.
Yes, exactly.
Eric asks:
is CRing somehow “smooth” with infinitesimal morphisms?
No, but see tangent category.
Right right. You were doing this work just as I switched jobs right at the beginning of the financial crisis. Self preservation required me to limit my time on the nCafe :)
Very neat stuff. I hope to spend some time on it after I get some more of the basics down. I’m still working on finite limits :)
You were doing this work […]
Actually, what I called “tangent categories” were just codomain fibrations and eventually it dawned on me that I was really needing the standard path fibration resolution that gives generalized universal bundles.
What is called tangent category at that entry is the codomain fibration followed by fiberwise abelianization . That’s an important extra step. It deserves the term “tangent” more, because that abelianization step is what mimics the linearization of the tangent space.
I am still not happy with that logics. Namely, for commutative algebraic monads, like in Nikolai Durov’s generalized geometry, the quasicoherent sheaves make something very close to Grothendieck topos (namely a vectoid which satisfies all the Giraud axioms for a Grothendieck topos but one which is therefore slightly weakenes), but this is not the abelian category. So, even in commutative situations, there is a more subtle operation then abelianization in general.
Does he get just a 1-category or is there an $(\infty,1)$-category hidden here? It seems to me that the logic “linearization = stabilization” makes sense only in the $(\infty,1)$-context. Abelianization of 1-categories is just a toy shadow that doesn’t capture it all.
Durov as I much discussed to you before has a homotopy theory of stacks in topoi, which gives an appropriate approach to his schemes. It may require some internalization, or enrichment to modify the usual $(\infty,1)$-categories but it is eventually compatible I believe. But you guys in infinity world should be interested in extension of the picture on the basis of such main examples, not just complaining why it is not already in your language.
I mean if Lurie has $(\infty,1)$-topoi, Durov generalizes topoi to vectoids then the next step may be generalizing $(\infty,1)$-topoi to $(\infty,1)$-vectoids. For example… :)
It deserves the term “tangent” more, because that abelianization step is what mimics the linearization of the tangent space.
I’m not even sure what you mean by “abelianization”, but whatever it is, I would guess it has more to do with taking some kind of “continuum limit” while linearization is a side effect. What is more fundamental? I need to thnk about it.
By the way, back in this discussion at the nCafe, you asked
what is a differential 1-form, really?
This is a question I’ve thought a lot about, but from the level of a caveman drawing on walls compared to what you’re doing. You offered two options:
Is a 1-form something that sends vectors to numbers?
Or is a 1-form something that sends vector fields to functions?
I would say neither. I think the duality should be between the geometric objects, not the corresponding dual vector spaces (which are continuum constructs and only really make sense in the continuum). To me, a 1-form is something that sends curves to numbers, a 2-form sends surfaces to numbers, etc. A curve could be defined as a continuum or some abstract finitary construction, but a 1-form does the same thing in either case. A 1-form lives to be integrated.
Have your thoughts changed much about 1-forms since then?
as I much discussed to you before
I know, we are going in circles here. You keep mentioning that this stuff exists, but I never do anything with it, nor do i watch anyone do anything with it. This way it never sticks.
You keep mentioning that this stuff exists, but I never do anything with it, nor do i watch anyone do anything with it.
Well it is up to you what you find interesting (and you certainly shown good feeling for many important stuff!), but I feel sad that after you did not find interest in any of my examples after 4-5 years of my trying, while I have spend lots of time and effort trying to understand yours and neglecting my own (so I am now in nobody’s land, being just second-hand and too slow for your stuff and simultaneously having lost too much time neglecting my own unfinished work).
Importance is by internal logical criterium and not by propaganda aspect. For example the geometry over a field of one element is supposed to give a natural proof of Riemann hypothesis. Similarly semiclassical expansion gives information about QFTs beyond the leading topological terms. These are examples of genuinly not understood phenomena what is more interesting to me than rephrasing the already known in more fancy terms. You see you enthusiastically claimed a number of times that cohomology in tersm of $(\infty,1)$-topoi gives all known cohomologies. When I give you candidates for counterexamples, you say I am not interested and have no time. Then you say FQFT is the way to unify all QFTs. Then I give you questions about (counter)examples where semiclassical expansion is not exact and you yawn again. Then you say Lurie’s point of view on generalized schemes is unifying all known “schemes”, then I give you examples of noncommutative schemes in terms of noncommutative localization, where we do not have Grothendieck topology but rather a Q-category or quasitopology in the sense of Rosenberg or noncommutative Grothendieck topology of van Oystaeyen. Again no interest. Then I tell you that I am really interested in your case of descent for non-groupoidal case (unorientals still agreeing with model structures as in nactwist) and you keep saying that all that work is superseded and hence not interesting to you anymore. I know it does not quite fit with concentration on $(\infty,1)$-world where you keep digging deeper and deeper and I wish I would have luxury more to dedicate with (with the ending of my contract in Dec, and travel semiban at my institute I can hardly make such a brave decision, and I apologize not to be able to fullfill our own hopes of going further with some of plans you sketched).
I do believe that slight but genuine extension of the picture we are developing in nlab could give natural framework to some of above mentioned examples, but one really needs to work with such “new” examples (if you do mainstream TQFT examples, then you naturally attract around people with such interests and after a while you see that everyone does such things; but this is the perspective of a shielded local cluster I think). Even very moderate results with genuinely new examples are often in my opinion (and for many people whom I know) worthy more than rephrasing known examples (what is also worthy and interesting, but does not compare). Not always of course, the understanding of the basics is sometimes more interesting; but the new applications seem to me further and further from the focus.
You see, you claim that 1-categorical abelianization is natural to look only from perspective of stabilization in $(\infty,1)$-context. I tell you that for the way you explain qcoh sheaves, you use abelianization where in general it does not make sense (as in general qcoh sheaves on generalized schemes do not form an abelian category). So even in 1-categorical situation there is something incomplete with that point of view. So why would somebody learn $(\infty,1)$-categories to get an incomplete point of view, which does not hold even for the natural class of 1-categorical geometries ? Unless one has ambition to go to the bottom, hence being concerned and try to solve the thing through. Thus I find my questions very logical and natural.
On a not unrelated note, did you see this question at MO, Zoran?
Thanks David, I will write my quick opinion there (I have to cook a dinner for a guest tonight so I will be quickly away).
Zoran,
here some replies:
in all this, time, I have been trying to understand nonabelian differential cohomology and nothing else. I adopted whatever seemed to be useful for that.
I am fond of some nice patterns that were unraveled on the way, yes. And I think a good pattern is worth more than a contrived counterexample. With many of the counterexamples that you present, I have the feeling that they break a nice insightful pattern without providing a similar benefit instead. In fact, in many cases it feels like with more work, they might fit into the nice pattern after all.
But even if not, the main point is to which extent it is useful. With many of these structures that you mention, I don’t know really know what their use is for the quest that I am on. I suppose probably they are relevant for understanding some non-commutative generalization, but then: I have little to no motivating examples for this.
I also wish we had had more interaction on these issues, but you can’t expect that if you mention some concept that sounds randomly picked to me, that I drop everything else and start studying that. I feel the progress that i did make in the past I made by following nice patterns, not by following curious irregularities in patterns.
I am sorry about how this probably sounds, but it’s the truth. How can I be intersted in something that I don’t know what it is good for. I am hoping that you can present some nice application of the work by Durov, Rosenberg etc. so that I would get a hook into the subject. I feel their work is too vast as that I invest lots of time into it without knowing what for and where it should lead.
Zoran,
concerning the notion of cohomology in an $(\infty,1)$-topos: I still don’t actually know what your counterexample is. I wanted to clarify this with you last time, because there is still a query box with a complaint of yours sitting in my $n$Lab entry, but I don’t thin kwe reached a joint conclusion. What I remember is that you started arguing that algebraic K-theory does not have a nice $\infty$-functorial formulation, but the counterargument was that on the contrary, Blumberg and Gepner gave a beautiful $(\infty,1)$-categorical formulaiton of all of K-theory.
So I am a bit at a loss as to what your complaint is about.
Zoran,
there are several points of what you write above where I don’t follow.
Here is one:
Then you say FQFT is the way to unify all QFTs. Then I give you questions about (counter)examples where semiclassical expansion is not exact and you yawn again.
I don’t know what that is about. That FQFT and AQFT are the two main paradigms for mathematical formalizaiton of quantum field theory is hardly my idea. There are plenty of examples of things that ought to be QFTs from the physics perspective, where we have neither the formal FQFT nor the formal AQfT formulation available. In fact, that’s the vast majority of examples. Still, the general feeling is that those examples where we do have the FQFT/AQFT formalization all understood strongly suggest that everything else should eventually fit into this.
I think math is about following patterns. When I see a pattern, I want to see as many examples as possible. If I have a bunch of them and then run into something that does not quite seem to fit, I don’t discard the nice pattern right away.
Urs, you have posted hundreds of answers to nearly random people say on cafe on noncommutative geometry, T-duality, motives etc. also out of scope for present day systematization in this effort. I do not feel that I was suggesting random examples. I am here suggesting repeatedly just 4-5 patterns steadily for 4-5 years. For example, we had an example of working few weeks in nlab on smoothness. The formal smoothness can be related to one of the principal examples of Q-categories, the same formalism which is used for one version of noncommutative spaces as sheaves. So it may have looked to you as two different questions but in bottom line it is the same question. Thus it is hard to believe that it is logical that you talk other aspects of nonommutativity e.g. noncommutative Dirac operators extensively e.g. on cafe, while the close basic question of finding appropriate way to look at noncommutative spaces as sheaves (not presheaves) is not motivating. I know that the average question by a superficial user of say nonommutative geometry coming accross your talks or across a discussion on the cafe will less likely ask questions on the foundations, but which and whose questions are more important ?
It is also not only a question of not fitting into pattern but the question of explaining genuinly new/unknown examples. People ask what is the main theorem and what are the main new examples covered and it is hard to answer such questions if the work centers on systematization. (In particular, in moments in which I tempted some propaganda among physicists, I have recieved pretty bad feedback on ncafe activity from theoretical physicists, many of which claimed to me missing understanding of physics there.)
You see, fitting syntomic cohomology into the pattern is trivial and it is basically waste of time, while fitting the extensions of nonabelian Hopf algebras is a genuinely nonabelian challenge (On the other hand, going into real specific properties of syntomic cohomology would be interesting, but a difficult subject out of the scope and our expertise).
In fact, in many cases it feels like with more work, they might fit into the nice pattern after all.
Right, but this work will bring, I believe things like some new extension of $(\infty,1)$-topoi etc. So instead of wasting huge time on claiming that generalized schemes in Lurie’s cover all examples, one really covers some really new examples with them or with an extension. I feel that I lost 2-3 years at least in learning the new techniques in order to stay now alone being concerned about doing really new applications, particularly those where I know something to start with (it is of no use to me to compete with american school of quasicategories and prove theorems on quasicategories, or the straightforward examples of those, I am not expert enough for that).
In particular, I see departure from real physics. I was hoping that the main concern would be to study the simplest effects beyond the topological term using still higher categorical methods. So eventually this could make all that TQFT and categorical stuff relevant for the study of typical QFTs. This I find extremely important.
Have your thoughts changed much about 1-forms since then?
Yes. I see they did change :)
A 1-form is a smooth functor $\mathcal{P}_1(X)\to \mathbf{B}G$. But instead of integration, it is product integration. Very pretty. I’ve glanced at these papers, but was too busy to absorb them back then.
If Toby was a coauthor, he probably would have insisted you start with 0-forms or maybe even (-1)-forms :)
Now I hope to understand how 2-groupoids work. This is a case where shapes begin to matter. My preferred shape would be a 2-simplex or a 2-diamond and probably not a 2-bigon.
By the way, a very slight critique. Why the emphasis on the continuum? It seems everything you did could be easily generalized to any kind of category and looking at Lie stuff is actually a very small corner of the possible universe of ideas. A corner that could easily be obtained at the end of the day by taking the continuum limit (or abelianization) of the more general ideas. I suspect this might not be totally unrelated to what Zoran wants to do.
The somewhat cludginess of stopping points (come on, admit it is cludgy) is a hint that nature cannot be like that. [Edit: Just in case the tone doesn’t come across correctly, I mean that last comment in a lighthearted manner :)]
Thus it is hard to believe that it is logical that you talk other aspects of nonommutativity e.g. noncommutative Dirac operators extensively e.g. on cafe, while the close basic question of finding appropriate way to look at noncommutative spaces as sheaves (not presheaves) is not motivating.
Here is the logic behind that: I think of spectral triples and their generalizations as being the algebraic encoding of certian FQFTs. This is what Stolz-Teichner and Soibelman and others have been accumulating evidence for. Specifically the Dirac-type operator is the infinitesimal generator of what the FQFT assigns to 1d cobordisms of Riemannian length $\ell$, as $\ell$ increases.
That gives a pretty nice picture of Connes-style noncommutative geometry, I think. Soibelman should be in the process of writing up a piece on that for our book.
What I personally don’t have good motivating examples for is the kind of noncommutative sheaves that you mention. Probably there are some. What are they?
I am not sure, it sounds a bit to me as if you are complaining that I didn ’t work out anything with that. But did you work out anyting with that? It seems a bit like you are trying to push me to do something that you might push yourself to do. You seem to have the motivation to do it, I am lacking it. I am pretty mjuch over-busy with what I am doing already.
I can understand if you are frustrated. I am frustrated, too. But I think the attitude that somehow it is all my fault is questionable.
Have your thoughts changed much about 1-forms since then?
I have a more general picture of what differential forms are. This I describe in the section Intrinsic de Rham cohomology.
(Warning: requires the notion of finite $(\infty,1)$-limits.)
Zoran,
just to open a different perspective: it so happens that I have by now a good understanding of all the topics in the nactwist notes if I assume that the context is a “locally oo-connected (oo,1)-topos”, such as oo-stacks on Cartesian spaces. In this context I can fully formalize and work out most or all of what we tried to formalize and work out back then.
What I find, in turn, a bit frustrating that instead of joining in this partial success, I see you complain about the assumptions being made. But this setup is something that works for what we did back then, so i am inclined to stick to it for a bit.
And if non-commutative generalizations should be called for, I am accordingly inclined to see if they can’t be realized in terms of ordinary oo-stacks on some site of noncommutative spaces. Because in that case I would then know how to proceed with studying what i want to study.
What I personally don’t have good motivating examples for is the kind of noncommutative sheaves that you mention. Probably there are some. What are they?
I can again answer such a question, if you are really interested. Just to give a hint, a manifold has open sets which generate topology, and for noncommutative algebraic spaces, the open sets are replaced by localizations. Localizations simply do not form a topology, not 1-topology, not infty-topology. The sheaves for such a topology do not make a topos, not 1-topos, not infty-topos. The examples of noncommutative localizations are many. Also noncommutative smooth topology is not a topology either in the same vain.
One more clarification for a misunderstanding in 26:
But did you work out anyting with that? It seems a bit like you are trying to push me to do something that you might push yourself to do.
I find it natural to discuss with you and others things which belong between the “expertise” of yours and of mine, and where one needs both (I don’t think I am really an expert on anything, but say so). I see no need to do (unless asked) the exposition of work I have already finished, or can do completely myself, nor on the contrary to delve completely into the things where you do not need my insight, and even where my still low level must be annoying and burdensome.
As far as 4-5 main mentioned questions, yes, I did spend lots of time on most of those, but mainly before I was aware of new techniques and points of view, which now make me more hopefull into resolution of those. Those new points of view include the stuff which I learned from this n-community and Igor and you especially, so my hopes are directed toward my teachers in new area of my study; hence it is a bit hard to take mainly negative response in most of the cases (like that I do not understand that the derived functor is just a way to raise something to a weak infinity-functor. Well, in some cases I know of “derived” where I suspect the matter may be a bit different and seek a resolution of this different than just relearning the nPOV better.)
For example of such things I keep mentioning and worked on, is about Hopf algebra extensions I have spent entire summer of 1997 and improvised hundreds of partial formulas in low dimension, but now I hope that some sort of nonabelian Schreier’s theory can be found by using a systematic approach in nonabelian homological algebra. Thus, it is rather crucial to know if the nonabelian homological algebra a la Bourn-Janelidze etc. or the non-symmetric recent version of Rosenberg give or give not derived functors in the sense of $(\infty,1)$-categories. Mike thinks that for semiabelian setup at least one could even find a canonical model category structure giving the same derived functors: we looked at some references but it is unconclusive in my understanding. For the right/left exact categories of Rosenberg the thing is less hopeful as in a way it has just half of the structure, that is structure for enabling the theory of either left or right derived functors. Now this theory is very simple, I mean just a choice of Grothendieck pretopology by strict epimorphisms. This setup reminds me pretty much of David Roberts’ thesis discussed here. Thus there are so many situations where I see the connection and raise a discussion. I am not a member of cafe to make long standalone expositions, but in topics of others I see connection to the way I think.
I will be probably offline most of the weekend, starting tomorrow, but maybe I manage to connect somehow and do some nlab work.
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