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This Friday in Halifax at the CMS meeting 2013 in the session organized by Mike Shulman, I’ll be giving a brief talk titled “Synthetic quantum field theory”.
I am still working on my pdf slides, which are supposed to contain some background information beyond what I’ll actually show in the talk itself. But that talk-part of the talk has been stabilizing now. You can see it as the beginning of this file, on the first 15 pages/29 slides:
I’ll be further finalizing this tomorrow, in some spare minutes…
With regards to Hilbert, have you seen Leo Corry’s On the origins of Hilbert’s sixth problem: physics and the empiricist approach to axiomatization?
I’m still a little uncertain how Hilbert viewed the comparative status of number theory, geometry and physics. In 1899 he said
Geometry also [like mechanics] emerges from the observation of nature, from experience. To this extent, it is an experimental science.…But its experimental foundations are so irrefutably and so generally acknowledged, they have been conﬁrmed to such a degree, that no further proof of them is deemed necessary. Moreover, all that is needed is to derive these foundations from a minimal set of independent axioms and thus to construct the whole ediﬁce of geometry by purely logical means. In this way [i.e., by means of the axiomatic treatment] geometry is turned into a pure mathematical science. In mechanics it is also the case that all physicists recognize its most basic facts. But the arrangement of the basic concepts is still subject to changes in perception …and therefore mechanics cannot yet be described today as a pure mathematical discipline, at least to the same extent that geometry is. (Hilbert, Grundlagen der Geometrie)
This suggests that geometry and physics (mechanics) have a similar status, although geometry at the time was further advanced. It seems these two differ from number theory:
Geometry is the science dealing with the properties of space. It differs essentially from pure mathematical domains such as the theory of numbers, algebra, or the theory of functions. The results of the latter are obtained through pure thinking …The situation is completely different in the case of geometry. I can never penetrate the properties of space by pure reﬂection, much the same as I can never recognize the basic laws of mechanics, the law of gravitation or any other physical law in this way. Space is not a product of my reﬂections. Rather, it is given to me through the senses.
I wonder what Hilbert would have made of the attempt to turn number theory into geometry, so that Manin could say of his image combining the ordinary geometric (odd and even) and the arithmetic ’dimensions’ that
The message of the picture is intended to be the following metaphysics underlying certain recent developments in geometry: all three types of geometric dimensions are on an equal footing.
organized by Mike Shulman
and Peter Lumsdaine.
I suppose I should probably have posted about this session at the n-Cafe or something.
Yes, any commentary on what’s happening would be welcome.
David,
in reply to your #2
yes, thanks, I have seen Corry’s text. The related references that I looked at are those listed at Hilbert’s 6th problem. But I haven’t read every single word, so further pointers are welcome. I should maybe include a citation from Corry’s abstract in my slides, for emphasis.
This suggests that geometry and physics (mechanics) have a similar status, although geometry at the time was further advanced.
Yes, I seem to understand that this is one of the points that Corry is making about Hilbert’s perspective.
It seems these two differ from number theory […] I wonder what Hilbert would have made of the attempt to turn number theory into geometry,
Yes, so I think nowadays this is hardly just an “attempt” anymore, and Manin is just one example of its proponents: by way of algebraic geometry the bulk of number theory is absorbed into geometry/topos theory, I’d say. For example Fermat’s last theorem was proven via geometry (elliptic curves).
I think with “geometry” understood as part of topos theory and in view notably of Univalent Foundations the qualitative distinction between geometry and number theory that apparently Hilbert perceived, as witnessed by the above quote, disappears.
By the way, concerning Manin’s triple-dimensions: I never quite understood why it makes sense to conflate the coefficients and the generators. To me the interesting triple of dimensionalities is rather
even non-nilpotent
odd nilpotent
even nilpotent
which are the three dimensionalities of formal supermanifolds.
But I may well be missing Manin’s point, of course.
Ah, I didn’t know about this triple you mention.
As far as Manin’s account goes, Lieven Le Bruyn gives some indications in Section 1.3 Manin’s geometric axis of his absolute geometry that (spec(Z[x])) can be seen as mixing the geometric and arithmetic, from which we can then project out in both directions:
Yuri I. Manin is advocating for years the point that we should take the terminology ’arithmetic surface’ for spec(Z[x]) a lot more seriously. That is, there ought to be, apart from the projection onto the ’z-axis’ (that is, the arithmetic axis spec(Z)) also a projection onto the ’x-axis’ which he calls the ’geometric axis’.
This appararently points in the direction of $F_1$.
Mike #3,
right, sorry, should have said so.
David #4,
here is a lightning summary of some of the things we heard yesterday:
Emily Riehl (with Dominic Verity) studies the homotopy 2-category of quasi-categories and observes that $\infty$-(co-)limits and $\infty$-adjunctions of $\infty$-categories can usefully be described and studied there. ($\infty$-adjunctions don’t involve higher coherences…)
(They refer to this as “formal quasi-category theory” in analogy with Gray’s famous term “formal category theory”, but I’d think one might want to use that term rather for a discussion of the full $(\infty,2)$-category of $\infty$-categories. )
Guillaume Brunerie presents the type-theoretic formulation of what Maltsiniotis a while back identified as Grothendieck’s hidden definition of weak globular $\infty$-groupoids.
Here it remained a bit unclear to me what the punchline was, in particular since the homotopy theory of these things is unknown. Probably Mike can say more.
André Joyal gave an introduction to the categorical semantics of homotopy type theory of the kind we have been discussing here at some length a while back.
Dimitri Ara reviewed his work on the model for $(\infty,n)$-categories by the model structure on cellular sets.
David #6,
I certainly see the point of taking seriously $Spec(\mathbb{Z}[x])$ as a geometric object. I just don’t see in which sense $Spec(\mathbb{Z})$ is an “axis” (“a dimension”) of it.
1: impressive!
Urs #8,
So the inclusion $\mathbb{Z} \to \mathbb{Z}[x]$, or projection $Spec(\mathbb{Z}[x]) \to Spec(\mathbb{Z})$ doesn’t stike you as resembling a projection to an axis? I’m not sure what would be expected either way.
David,
yeah, so that inclusion $\mathbb{Z} \to \mathbb{Z}[x]$ (embed constant functions) is dually the map from the “curve” to the point, and the projection $\mathbb{Z}[x] \to \mathbb{Z}$ (evaluation of functions at zero!) is dually the map which includes the point as the origin of the “curve”.
It might help to think of this for the coefficients $\mathbb{Z}$ replaced by $\mathbb{R}$ and for free rings replaced by free smooth algebras. Then the projection is $ev_0 : C^\infty(\mathbb{R}) \to C^\infty(\ast) = \mathbb{R}$, which is dual to $0 \colon \ast \to \mathbb{R}$.
That’s why I am saying: I am not sure in which sense it is supposed to be useful to regard the coefficients as a “dimension”. Ordinarily the two play different roles, and jointly give a notion of space namely of some dimension and over some base.
Or in other words, the choice of coefficients is certainly a “degree of freedom” when defining a geometry, but it is a different kind of degree of freedom than the choice of dimension.
Zoran #9,
thanks. Let me know if you spot something you’d like particularly to see improved, then I can still edit it.
@Urs
Surely it’s more straightforward than that: $\operatorname{Spec} \mathbb{Z}$ is a one-dimensional regular affine curve, literally! Similarly, $\operatorname{Spec} \mathbb{Z}[x]$ is a two-dimensional regular affine scheme.
Zhen Lin,
the analog of what I am complaining about is:
a) the point is a 0-dimensional manifold.
b) the line is a 1-dimensional manifold.
c) but the inclusion of the point in the line is not a “coordinate axis”.
But let’s reverse this discussion, to make it more fruitful: I am wondering what is Manin’s idea of how in
$\mathbb{Z}[x_1, \cdots x_p, \xi_1, \cdots \xi_q]$the $\mathbb{Z}$ corresponds to “a kind of dimension” on the same footing as the $x_i$s and the $\xi_j$-s do?
I may be missing the point here. If you know the answer, let me know.
But $Spec \mathbb{Z}$ is not a point! (Who says terminal objects have to be pointlike?) The idea, possibly going back to the early 1900s, is that $\mathbb{Z}$ is (the coordinate ring of) a smooth curve like the line, whose points are the various primes. The “arithmetic surface” $Spec \mathbb{Z}[x]$ is then thought of as being fibred over $Spec \mathbb{Z}$, with fibres the affine lines over the various prime fields. The comparison to $Spec \mathbb{C}$ and $Spec \mathbb{C}[x]$ is misleading and should be avoided: the spectrum of an algebraically closed field is indeed just a simple point.
If you look at Zariski spectra then already every $C(X, \mathbb{C})$ for $X$ a manifold gives wild spaces with loads of extra points. If Zariski spectra is what Manin has in mind, then he’d need to add plenty and plenty of spurious “new dimensions” even for ordinary supergeometry.
But I think I’d rather try to find Manin’s original article myself and have us get back on topic here.
Who said anything about manifolds or supergeometry? For rings like $C(X, \mathbb{C})$, obviously one uses a different spectrum – but even the simple-minded $MaxSpec$ works for compact $X$. And $MaxSpec \mathbb{Z}$ is still very much infinite and far from being a point.
Who said anything about manifolds or supergeometry?
Re: 7, one thing I find interesting about Guillaume’s work is that it shows that the notion of ’synthetic $\infty$-groupoid’ studied by HoTT really is nearly the same as Grothendieck’s original proposed definition of $\infty$-groupoid. That Grothendieck knew what he was doing.
More concretely, it’s part of the search for a more computational understanding of HoTT. One approach to the latter is to build the $\infty$-groupoid operations into the type theory, and that seems more tractable with a nice syntactic presentation of what it means to be an $\infty$-groupoid.
Thanks, Mike.
I guess what I am wondering about is what it is that has been achieved as long as only the globular $\infty$-groupoids have been defined, without having a good notion of maps between them.
Of course I entirely agree that the definition looks good. But to establish this and to rule out the possibility that the evident intuition is wrong after all, isn’t it urgent that one first thinks about producing the correct “mapping types” between these globular $\infty$-groupoids? Or do we learn about the points that you mention (e.g. computational understanding) already without these?
And finally I am wondering: given the slick nature of the definition of these globular weak $\infty$-groupoids, is there not maybe now a similarly slick definition of the weak maps? Maybe by proceeding in the same spirit and saying something like “for every contracticle pasting diagram in the domain there is a contractible pasting diagram in the codomain such that…”
Well, even if your end goal is to define a category, you generally have to define the objects before you can even think about defining the maps. If both are difficult in some case, then succeeding at the first is an achievement, whether or not you’ve done the second yet. Of course the second is the next step, but I find it funny to describe anything in abstract pure mathematics as “urgent”.
The same techniques to define weak maps that work for Batanin structures – strict maps out of cofibrant replacements – can be used with Grothendieck ones, of course, and your suggestion is also quite natural. You could talk to Guillaume about it while you’re both still here in Halifax!
@David,
thanks for catching typos! Fixed now.
@Mike,
thanks, I see. Yes, I’ll try to talk to Guillaume (and to you)! As usual,there is always too much going on at once… (That reminds me: for booking-technical reasons I’ll be here also on Saturday…)
If Zariski spectra is what Manin has in mind, then he’d need to add plenty and plenty of spurious “new dimensions” even for ordinary supergeometry.
Rosenberg has many times pointed out that while a scheme (say, as a ringed space, with mild conditions on the scheme) can be reconstructed from the abelian category of quasicoherent sheaves, his definition of a spectrum of abelian category gives much more points to the superscheme, then the spectrum of the underlying commutative scheme has (also, as I noted independently, the localizations of the category of quasicoherent sheaves give somewhat finer Zariski topology for superschemes, even if one does not care about points, and having better Zariski topology is always a plus!!). But he never wrote more details on this, and I do not know anybody who followed that idea (pronounced e.g. in his 1995 book). I thought of this idea for my category of quasicoherent sheaves on certain generalization of schemes motivated by the problem of integration of Leibniz algebras, but did not get far there as I thought to little on this so far.
@Zoran,
I think the point here is that generally Zariski-like topologies are useful for turning number theory and algebra into geometric language, but are fairly alien to the kind of physics-inspired geometry that Hilbert would have thought of, which is what usually goes by “differential geometry” and related.
But nowadays it’s pretty standard that the correct route for applying algebraic-geometry-style reasoning to differential geometry is not to fiddle with the ill-suited Zariski topology on ordinary rings, but to pass altogether from the theory of plain rings to that of algebras over a suitable Lawvere theory, for instance from locally ringed spaces to locally algebra-ed toposes, for instance for smooth algebras (“$C^\infty$-rings”) or their supergeometric analog, in the case at hand.
@David,
thanks! Fixed. And expanded a little more.
(Will be further editing in two hours. Now first some coffee, then Thomas Nikolaus’ talk!)
26: the phenomenon I am talking is NOT restricted to Zariski topology, but it is the easiest to spot there. For any topology, you will get also covers which are kind of skew. That is not covering on commutative spectrum with additional conditions and or structure, but somehow you have some “open sets” which whose description is a mixture of odd and even coordinates. This certainly has repercussions to physics context. On the other hand, if you want physics, you can also imagine generalized GAGA principles to get more functional spaces. In any case, algebraic functions are dense in whatever functional spaces you want to add later and the spectral phenomena at algebraic heart are precursors of spectral phenomena at functional level and sometimes there is even full correspondence.
Mike,
so I finally talked with Guillaume, and here is something I find fascinating:
he says it’s straightforward to generalize his definition of (what looks like) globular $\infty$-groupoids to something that looks like globular $(\infty,1)$-categories, simply by enforcing an ordering condition on the 1-cells in the pasting diagrams that enter the definition.
If that holds water, it might be a way to circumvent that problem with defining semi-Segal types, and in a rather nice way.
Of course one still needs to sort out the maps between these structures…
Ah, here is something I forgot to ask: I suppose it’s easy to say when a strict map of Brunerie-$\infty$-groupoids is a weak homotopy equivalence? If that is the case then one could at least start talking about $\infty$-anafunctors beween them in what should be a straightforward way.
Regarding Julie Bergner’s talk, what examples are there of the phenomenon that
Many naturally-arising $(\infty,1)$-categories are equipped with a group action.
After the lobster dinner André Joyal had kindly explained to me the $\infty$-topos of parameterized spectra that he talked about at IAS a while back.
I suppose one way to state it (though that is not verbatim how André described it, and so if the following is wrong that’s entirely my fault) is to say that it is the $(\infty,1)$-Grothendieck construction of the $\infty$-functor
$Func(-,Spectra) \colon \infty Grpd^{op} \to (\infty,1)Cat \,.$He also says that this is a cohesive $\infty$-topos. I need to think about this…
Yes, one could certainly hope that Guillaume’s definition of higher categories/groupoids could be implemented in type theory, but I suspect that it will run into similar sorts of problems as any other approach. However, I would love to be proven wrong.
One should be able to define weak equivalences in the same way as for other algebraic definitions.
Julie’s talk was a little thin on examples, but she mentioned the orthogonal group actions that arise in Lurie’s cobordism hypothesis theory.
And yes, I think that’s a good definition of the topos of parametrized spectra. I still haven’t really internalized the idea that this is a topos, but it’s an exciting idea.
she mentioned the orthogonal group actions that arise in Lurie’s cobordism hypothesis theory
I didn’t think that there was an open issue in how they are defined, though…
the idea that this is a topos, but it’s an exciting idea.
Yes, it’s interesting in how simple it makes, or would make, the internal description of spectra. I certainly didn’t appreciate the idea before that an $\infty$-topos can contain stable objects in the first place.
Also I bet that for $\infty$-toposes $\mathbf{H}$ with sufficiently good shape modality $\int$, also $Func(\int(-),Spectra) : \mathbf{H}^{op} \to Spectra$ induces an $\infty$-topos… But I haven’t really thought about it yet.
Re Urs #32
[Joyal] also says that this is a cohesive $\infty$-topos.
So there’s a new kind of thickened point here?
So there’s a new kind of thickened point here?
Yes, definitely.
(Assuming the claim is true, which I have no reason to doubt, but I also haven’t convinced myself of it yet.)
And it should be interesting, something like the “abstract stable point”, whatever that turns out to mean. I want to understand this better. This should be a rather important example. I am glad this has surfaced. In fact I think I was very lucky to have had a chance to talk with André Joyal.
I think Andre and Eric Finster said they thought this was the classifying topos of ’stable objects’ (objects that are the loop space of their suspension), with the generic stable object being the sphere spectrum over a point.
I didn’t realize it was cohesive, though. Off the top of my head, it seems like the discrete and codiscrete objects should coincide, both being the category of spaces with their parametrized zero spectrum (since the zero spectrum is both initial and terminal). Is that possible?
the “abstract stable point”
So after that discussion on Kapranov’s talk Categorification of supersymmetry and stable homotopy groups of spheres, is there a sense in which the abstract stable point is to the superpoint as the free abelian ∞-group on a single generator (the sphere spectrum) is to the free abelian 2-group on a single generator?
it seems like the discrete and codiscrete objects should coincide, both being the category of spaces with their parametrized zero spectrum (since the zero spectrum is both initial and terminal). Is that possible?
I still need to better understand this $\infty$-topos of Joyal.
I may be wrong, but I started to think that it is actually the tangent (infinity,1)-category to $\infty$-groupoids
$T_{\infty Grpd} \coloneqq FiberwiseStab\left( \infty Grpd^{\Delta^1} \to \infty Grpd \right) \,.$I haven’t checked this fully formally, but it seems to be clear.
Now the unstabilized codomain fibration $\infty Grpd^{\Delta^1} \to \infty Grpd$ is the shape functor of a cohesive adjoint quadruple
$\infty Grpd^{\Delta^1} \to \infty Grpd$which regards a bundle of $\infty$-groupoids $P \to X$ as being an $X$-parameterized collection of cohesively contractible spaces $P_x$, in that the shape modality takes $P \to X$ to $X \to X$, while the flat modality takes it to $P \to P$.
Judging from this, I feel like expecting that the cohesion on the Joyal $\infty$-topos is such that the shape modality takes a bundle of spectra to the 0-bundle on the underlying homotopy type, and that the flat modality takes it to the 0-bundle on $\Omega^\infty$ of its total space.
But this is just a guess, I haven’t checked.
So are we entering Goodwillie calculus territory? I seem to recall something about spectra as forming a tangent space, ah yes here. Oh, so it’s the same Eric Finster recently mentioned as working on opetopic type theory. He wrote a blog post here explaining some of his ideas, including
One tantalizing aspect of the Goodwillie calculus is that it suggests the possibility of thinking geometrically about the global structure of homotopy theory. In this interpretation, the category of spectra plays the role of the tangent space to the category of spaces at the one-point space. Moreover, the identity functor from spaces to spaces is not linear…and one can interpret this as saying that spaces have some kind of non-trivial curvature.
Maybe that thought above about the superpoint being a truncation of the abstract stable point isn’t so off target.
EDIT: Of course, you knew about this already Urs, as there you are commenting on Eric’s post.
My somewhat side-tracked comment in #39 nothwihstanding, Mike of course seems to be right in his #37.
I’ll now start a note on this tangent cohesion somewhere on the $n$Lab to provide a place to take better stock of what we understand and what we don’t understand yet.
Let’s continue discussion this here.
I am now adding “Example 3” to the “Synthetic quantum field theory”-slides which is the brane bouquet-story.
Still needs a good bit of polishing, but if anyone feels like a “lighthouse customer” ;-), then check out from page 64 on here.
Re: the original post in this thread, everyone’s slides from the special session are now available here.
Elsewhere David C. suggests to look at
for another quote which might usefully close the conclusion section of “Synthetic QFT”.
While browsing through the text, I’ll extract some quotes that seem noteworthy, just so as not to forget them in case they later turn out to be useful:
p. 392 (405) :
Ja es hat ganz den Anschein, dass ähnlich wie in der Einsteinschen Gravitationstheorie eine weitere Vertiefung und Ausbildung dieser mathematischen Methoden später einmal nötig ist, um in befriedigender Weise die Erklärung und Beschreibung jener noch recht im Dunkel liegenden Einzelvorgänge zu bewirken.
from p. 402 (414) on:
interesting how Hilbert reviews – from the point of view of having known the right answer all along from the variational principle (hence from the axioms…) – Einstein’s struggle (until a few days before the lecture was given!) with finding the right equations of gravity This comment culminates on p. 405 (418) with the curious remark, paraphrasing just a little: that the fact that Einstein after his “colossal detour” ends up with the same equations of motion as Hilbert already had is a “nice consistency check” (schöne Gewähr)
Heh.
p. 417 (430):
Hilbert almost finds himself coming to agreement with Hegel…
Dieses Ergebnis scheint uns fast auf den Hegelschen Standpunkt zu führen, wonach aus blossen Begriffen alle Beschaffenheit der Natur rein logisch deduziert werden kann. Aber bei näherem Zusehen kommen wir zu einem Standpunkt, der demjenigen von Hegel vielmehr ganz entgegengesetzt ist. Wir stehen da vor der Entscheidung über ein wichtiges philosophisches Problem, nämlich vor der alten Frage nach dem Anteil, den das Denken einerseits und die Erfahrung andererseits an unserer Erkenntnis haben. Diese alte Frage ist berechtigt; denn sie beantworten, heisst im Grunde feststellen, welcher Art unsere naturwissenschaftliche Erkenntnis überhaupt ist und in welchem Sinne das Wissen das wir in dem naturwissenschaftlichen Betriebe sammeln, Wahrheit ist.
…only to then obtain the opposite conclusion after all:
p. 424 (437):
fassen wir den gesammten phys. Wissenkomplex ins Auge und fragen nach dem Begriffsfachwerk dieses Gesammtwissens und nach den Axiomen, auf denen dieses grösste, Alles umfassende Fachwerk beruht. Meine Antwort lautet: unsere Weltgleichungen sind diese Axiome und das volle System aller math. Folgerungenaus den Weltgleichungen bildet das Fachwerk des phys. Wissenskomplexes, so dass dann der math. und begriffliche Ausbau der Weltgl. im Prinzip alle phys. Theorieen enthalten muss. — Wenn nun diese Weltgleichungen und damit das Fachwerk vollständig vorläge, und wir wüssten, dass es auf die Wirklichkeit in ihrer Gesamtheit passt und dann bedarf es tatsächlich nur des Denkens d. h. der begrifflichen Deduktion, um alles phys. Wissen zu gewinnen; als dann hätte Hegel Recht mit der Behauptung, alles Naturgeschehen aus Begriffen deduziren zu können. […] Im Gegensatz zu Hegel behaupte ich, dass gerade dieWeltgesetze118 auf keine andereWeise zu gewinnen sind, als aus der Erfahrung.
Hm, so I guess I’d rather be citing Hegel, then… :-)
There are quite a few steps between an empiricist and a Hegelian (or objective idealist). The former’s view of scientific theory generally is that it’s an efficient encoding of the observations we make of the world, accompanied typically by a wariness about taking the theory too seriously, e.g., as to what it says about the unobservable. Observations, on the other hand, are fairly straightforwardly given to us by the world. Science’s basic task is to codify them, so that we can predict and sometimes control nature.
Germany had for many years gone down a non-empiricist path. Kant had wanted to know, given the existence of mathematics and Newtonian theory, what must be the case about our cognitive faculties. He worked out that we impose certain structures in our construction of experience, time and space, causality, etc. No simple empiricism then. We can’t see the world as it is in itself. He read off twelve “categories” from the types of judgement we make.
Hegel criticises him for only listing the 12 categories. The thing was to deduce them from first principles. What could come first? A Logic of the Idea. We starting with Being and Nothingness, note their identity and simultaneously their difference, deduce Becoming, and off we go, through a large number of twists and turns to explain why the world must be as it is.
The world is secondary. The Idea has an internal dynamic which is driven by the dialectical process. It requires a world to play itself out in. We are a vehicle for the Idea.
For Hegel, there’s no Kantian separation between us and the thing-in-itself, unexperienced through our faculties. We and our experience are just one part of the working out of the Idea.
You’re most like Hegel when you’re doing your HoTT analysis of physics. It has the flavour of working out how the world must be for structural reasons.
Thanks David.
I do remember that you told me this before, but what you just wrote is very nicely put. To record it, I have taken the liberty of pasting your comment into a new $n$Lab entry objective idealism.
Now I feel very uneducated for asking the following, but I’ll ask nevertheless: could you give me a pointer to a specific volume and page number(s) where I might see Hegel say all this more or less explicitly? Or does he never condense it to a few paragraphs like this?
If ever you want a pair of opposites, pick ’Hegel’ and ’conciseness’. And they don’t form one of his unity of opposites, so beloved by Lawvere.
His ’shorter’ Logic is already very long, and still in places the arguments are rather abbreviated. I very much doubt it’s in the details of his reasoning that one finds inspiration, but rather in a general attitude. He’s enormously hampered by a lack of decent logical apparatus. That’s what Lawvere thinks category theory can supply, e.g., adjunctions as unities of opposites. I don’t know whether Lawvere thinks that one can encode the smallest details of Hegel’s Logic this way, or whether it’s more of a general inspiration.
Hegel would probably be pushing you on to explain why it’s inevitable that the four stages of your slide 9 appear as they do.
Hi David,
okay, I sort of expected that, it’s a pity. Then I’ll just quote your summary of Hegel. :-)
Concerning your suggestion on motivating the four axioms of synthetic QFT: yes, I have wondered about that, too. Asking myself that kind of question is for instance what made me highlight the I Ching-structure of the modalities on p. 35, slide 53.
first axiom: not really an axiom at all, it just says to use foundational logic;
second axiom: is canonical in the following sense: given the fondational logic, the remaining freedom is adding modalities. A strong version of these is adjoint modalities. About the strongest of these that still admits interesting models is adjoint triples of modalities. There are precisely two choices for such: the yin-triple: monad-comonad-monad and the yang-triple: comonad-monad-comonad. So take them both. That’s the axiom of differential cohesion.
third axiom: not really an axiom either, rather the advise: use the canonical structures induced by the previous axiom: consider the homotopy fibers of the units of the comonads and slice over them (equivalently: make other types dependent on them). And consider relations in these slices. That’s synthetic prequantum field theory.
fourth axiom: that’s the one where there is certainly still the most room to understand how it is fully “god-given”. This is why I was after understanding the deep universal meaning of “motivic” stuff lately. But even at the not-yet-super-deep-level at which I have it currently, I think it’s looking pretty canonical. It says: linearize the above relations in the slice and sum them up. Eventually I hope this will be a canonical universal left adjoint construction on “relations dependent on $E_\infty$-ring types”. But this is the point I don’t fully understand yet at the Hegelian level.
Since we’re in philosophical mood, have we learned anything more about that Kantian question that has cropped up from time to time, concerning the degree to which our mathematics reflects our shaping of experience? In statistical mechanics we seem more ready to attribute parts of the mathematics to the inference process. But then quantum mechanics is just a Wick rotation away, as here:
I guess what Urs and I are wondering is what to make of this fact. Having lectured statistical mechanics to our students, do we say to them “don’t bother going now to the quantum mechanics lecture, you can just change the rig in what I told you.”?
Obviously we don’t do that — even though we could. Instead, we first teach them quantum mechanics, and later we say something like this:
There’s a deep, mysterious relation between quantum mechanics and statistical mechanics, called ‘Wick rotation’, which consists of replacing time by imaginary time. This lets us apply all the techniques of statistical mechanics to quantum mechanics. It’s especially useful in quantum field theory — especially for understanding concepts like ‘renormalization’, ‘phase transitions’, ‘critical points’, ‘spontaneous symmetry breaking’, and so on.
Deep questions. Two thoughts:
for more formally connecting “Synthetic QFT” to our experience there might be need for a fifth axiom, which states how to assign a system of Bohr toposes to whatever axiom 4 spits out. This would be to be such that the internal logic of these Bohr toposes is the logic of observations in the given quantum systems. But I don’t have a good formal story for how to make this connection yet.
The relation between QFT and statistical physics is most pronounced in its incarnation as partition functions in QFT looking like expectation values in thermodynamics for states with “complex energy” values. Now the notion of partition functions at least is usefully recognized in the abstract axiomatics: these are just the genera that are given by the cohomological quantization of axiom 4.
For instance for the quantization of the superstring as push-forward in (twisted) tmf cohomology the result is the partition function which to a worldheet (“2d-spacetime”) torus of modulus $Z(e^{2\pi i \tau}) = sTr( e^{2 \pi \tau i L_0 - 2 \pi i \bar \tau \bar L_0} ) \\\tau$ in the upper half plane assigns
where $H_0 = L_0 + \bar L_0$ is the worldsheet Hamiltonian and $P_0 = L_0 - \bar L_0$ the spatial worldsheet momentum
(this assignment is the elliptic genus for the type II superstring and the Witten genus for the heterotic superstring).
Now the whole “Wick rotation magic” comes from these expressions being a priori obtained as functions on the moduli $\tau$ for elliptic curves, but as formulas being interpreted as functions on larger parts of the complex plane.
David Corfield (40):
So are we entering Goodwillie calculus territory?
Yes. In fact, for every $n$, the theory of polynomial functors of degree $n$ from spaces to spaces is an $\infty$-topos: this is an observation due to Georg Biedermann.
Joyal’s parameterized spectrum example is just the $n=1$ case.
Thanks for mentioning Biedermann, this way finally I found a reference for this: it appears in 35.5 of
Is there more, meanwhile? Has this appeared anywhere else?
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