Thanks for mentioning Biedermann, this way finally I found a reference for this: it appears in 35.5 of

- André Joyal,
*Notes on Logoi*, 2008 (pdf)

Is there more, meanwhile? Has this appeared anywhere else?

]]>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.

]]>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.

]]>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.

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.

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.

]]>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?

]]>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.

]]>Elsewhere David C. suggests to look at

- Sauer, Majer,
*David Hilbert’s “Lectures on the foundations of physics”*

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… :-)

]]>Re: the original post in this thread, everyone’s slides from the special session are now available 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.

]]>Let’s continue discussion this here.

]]>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.

]]>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.

]]>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.

]]>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?

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?

]]>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.

]]>Re Urs #32

[Joyal] also says that this is a cohesive $\infty$-topos.

So there’s a new kind of thickened point here?

]]>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.

]]>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.

]]>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…

]]>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.

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.