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Today at Spaces for Mathematics and Physics Maxim Kontsevich gave a survey of what the $n$Lab calls derived noncommutative geometry. That reminded me that I had long wanted to extract some more of the essence in a clear way.
The discussions I have seen about this, and today was no exception, have a lot of dg-stuff in it, and fall back to presentations when possible, and generally seem to be more interested in describing examples than in developing abstract theory.
But at least verbally there is the indication that what should really be going on is the following, and in saying this I allow myself freely to strip away the dg-ism and just speak $\infty$-categorically right away. Then in my words the suggestion that I hear is being made is the following.
We recall that
We may think of $\infty$-sheaf $\infty$-toposes with geometric morphisms between them as geometric spaces.
By the $\infty$-Giraud theorem these are (accessible) left exact reflections of $\infty$-catgegories of $\infty$-presheaves with values in $\infty$-groupoids.
The “stable $\infty$-Giraud theorem” (also by Lurie, of course) says that, analogously, every locally presentable stable $\infty$-category is the left exact localization of an $\infty$-category of presheaves of spectra.
Given that, it is entirely reasonable to ask whether one gets a sensible notion of geometry from the category of locally presentable stable $\infty$-categories with some suitable kind of geometric morphisms between them.
This is Maxim Kontsevich’s proposal, modulo the preference for stable $\infty$-categories which come from $H k$-module spectra, these are equivalently the $k$-linear enhanced dg-categories (by stable Dold-Kan correspondence).
Indeed, this perspective via the “stable Giraud theorem” makes it natural not to consider monoidal stable $\infty$-categories, which is the key point that is being advertized as making this be about non-commutative geometry (because the categories of $A_\infty$-modules over $A_\infty$-rings are not in general monoidal, and these serve as the affine spaces in this context).
The $n$Lab entry derived noncommutative geometry more or less says this already, but it could say it more clearly still, it seems to me.
But is this actually said in a clear way anywhere in the literature? I mean the abstract story. Or any abstract story.
There are several choices of what one would want to call geometric morphisms between stable $\infty$-categories, and maybe some care would be justified to lay these out a bit.
Today was suggested the definition that the $n$Lab calls the Grothendieck context (except for the monoidal structure, which is being ignored here, as I said). That’s clearly good for some things, but elsewhere one would want what the $n$Lab calls the Wirthmüller context (again without the monoidal structure). In fact that choice would connect the whole idea just seamlessly into the linear homotopy type theory story. Indeed, there the lack of the tensor product is quite natural: it just means that we start in a very weak fragment of linear type theory.
I am saying this in part just as a reminder to myself that, viewed this way, the story of Quantization via Linear homotopy types (schreiber) connects nicely – indeed it gives some conceptualization of what happens as we linearize those slice $\infty$-toposes to stable $\infty$-categories: we quantize, and indeed that’s where all the motivation for the Kontsevich style stable geometry comes from, the example of interest are the Fukaya- and $DQCoh$-categories that reflect the quantization of Calabi-Yau geometries as seen by quantum tological strings propagating on these.
Okay, that was just me mumbling to myself. Back to turning this here into a conversation: is there any place in the literature where an abstract “stable $\infty$-geometry” in direct analogy to “geoemtry as $\infty$-toposes with geometric morphisms between them” is laid out, or else discussed far enough that one may directly extract what should be the abstract $\infty$-categorical formulation?
This is a minor point, but not every presentable stable ∞-category is the stabilization of an ∞-topos.
Questions that I’d like to know the answer to include this:
in $\infty Grpd$, parameterized spectra have base change that satisfies Beck-Chevalley and projection formula. What are sufficient conditions for this to be true for parameterized spectrum objects in more general $\infty$-toposes?
the collection of all parameterized spectrum objects in an $\infty$-topos forms itself an $\infty$-tops. Conversely, what are sufficient conditions for a hyperdoctrine of stable $\infty$-categories over an $\infty$-topos to form an $\infty$-topos itself?
In addition to Marc’s point, even if stabilization is a natural thing to do, it’s not clear to me why one would expect this to have anything to do with noncommutative geometry. Also, it doesn’t explain why it is natural to work only up to Morita equivalence in noncommutative geometry.
On the other hand, if one takes associative ring spectra as a starting point, then their homological algebra essentially is the theory of (presentable) stable $\infty$-categories up to Morita equivalence. The latter being, of course, what is usually referred to as noncommutative algebraic geometry these days.
In addition to Marc’s point,
I don’t think that point is of concern for the question at hand.
You write
it’s not clear to me why one would expect this to have anything to do with noncommutative geometry
followed by
if one takes associative ring spectra as a starting point, then their homological algebra essentially is the theory of (presentable) stable ∞-categories up to Morita equivalence. The latter being, of course, what is usually referred to as noncommutative algebraic geometry these days.
Is the second statement not contradicting the first?
Let me try to say that again. You point out that $\infty$-toposes are roughly some kind of geometric spaces, and that it is natural to consider a stable version, i.e. left exact localizations of prestacks of spectra. However, why should I believe that this would give a good notion of noncommutative space? I can also propose other constructions that don’t make use of a symmetric monoidal product either, but have nothing to do with noncommutative geometry… Yes, every stable presentable $\infty$-category is a left-exact localization of an $\infty$-category of prestacks of spectra, so in retrospect one can also understand noncommutative spaces that way, but I don’t know any noncommutative spaces that arise in nature most naturally with a presentation of that form, do you?
The second statement was an alternative explanation for the appearance of stable presentable $\infty$-categories in noncommutative geometry, with the starting point of associative ring spectra, which are of course a reasonable thing to start with when trying to motivate a definition of noncommutative space. Namely, if $R$ is an associative ring spectrum, any $R$-algebra $A$ has an $R$-linear presentable stable $\infty$-category $Mod_A$ of right modules. These are precisely the categories which noncommutative algebraic geometry is concerned with. (In fact, one normally imposes the condition that the stable $\infty$-categories are saturated or smooth and proper, hence in particular admit a compact generator, hence are of the form $Mod_A$ by Schwede-Shipley.) In other words, “noncommutative algebraic geometry” is really just homological algebra of associative ring spectra or dg-algebras.
in ∞Grpd, parameterized spectra have base change that satisfies Beck-Chevalley and projection formula. What are sufficient conditions for this to be true for parameterized spectrum objects in more general ∞-toposes?
The base change and projection formulas are general properties of etale geometric morphisms, see Remarks 6.3.5.8 and 6.3.5.12 in HTT. The stable statements follow from these unstable ones.
why should I believe that this would give a good notion of noncommutative space? […] but I don’t know any noncommutative spaces that arise in nature most naturally with a presentation of that form, do you?
Maxim Kontsevich is the one who has been proposing since long time ago that a good notion of derived noncommutative geometry is given by enhanced dg-categories without tensor product and with certain adjoints between them as morphisms. Promted by hearing another talk where he made that point, justifying it by way of examples, I added some observations that serve to substantiate this more abstractly.
The second statement was an alternative explanation for the appearance of stable presentable ∞-categories in noncommutative geometry,
I understand, and therefore I was wondering, and still am, how you give this explanation and at the same time reject what it is explaining.
Marc, re #7, I’d be happy to see it all being formal, but I am not sure yet if your reply addresses the question.
For being more definite, consider an $E_\infty$-ring object internal to some $\infty$-topos and then then system of internal module spectra parameterized over objects of this $\infty$-topos. I’d like to know if this satisfies Beck-Chevalley and projection formula for its base change. I know how to prove this internal to $\infty Grpd$ using that left Kan extension may be defined pointwise over homotopy fibers. I wouldn’t be surprised but don’t know if this generalizes to arbitrary ambient $\infty$-toposes.
Suppose you have a cartesian square of ∞-topoi $f q=p g$ where $p$ is etale. Then $q_!g^*\to f^*p_!$ is an equivalence by HTT 6.3.5.8. I assume that’s what counts as Beck-Chevalley for parametrized spaces (or the mate equivalence between right adjoints). This certainly generalizes formally to spectra and then to modules over an $E_\infty$-ring (in the bottom right corner). One way to prove it is to observe that both upper stars and lower shrieks commute with the stabilization/free module functors.
Here is the video recording of the panel discussion of Topos à l’IHES last week in Paris.
At 57:50 André Joyal and then at 59:30 Alain Connes refer to the pre-quantum aspects of toposes, suggesting that true quantum geometry happens elsewhere. At 1.00:05 I am replying to this by (trying, maybe failing) to point out that it is precisely André’s tangent infinity-toposes which do accomodate genuine quantum geometry. This gives occasion, at 1:00:22, to highlight Maxim Kontsevich’s non-commutative geometry via stable infinity-categories. At 1:01:25 André passes this remark on to Maxim, at 1:01:38 Maxim comments himself.
Good to be chatting with Fields medallists, but not exactly a meeting of minds. I suppose the person who would best understand the tangent issue is not so versed in the quantum issue, and vice versa.
I am just relieved that I managed to squeeze in the comment and make the exchange with Maxim happen at the very end at all. One of the panel members used up an unproportional amount of the total time so that the intended discussion with the audience almost failed to happen.
How much of the discussion was in French? I only got to the very end of the panel, so it’s hard to know if it’s worth sitting through the video with my poor French or if I’d get most of it.
All contribution by Lafforgue is in French, which means close to half of the total time is. Joyal’s second contribution starts out in French, but soon he switches back to English.
Thanks, Urs ;-)
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