Perhaps the Dorn-Douglas approach to higher categories will be relevant here:

]]>Finally, one of my favorite and dearest parts of geometric higher categorical thinking (which, at the same time, is also one of the most mysterious parts for me) is the role that invertibility and dualizability plays in it. Here, of course, the cobordism hypothesis finally comes into play: invertible morphisms are modelled not by manifold diagrams, but by so-called tangle diagrams — (omitting all details,) tangle diagrams are simply a variation of manifold diagrams in which we allow strata to ‘change directions’ with respect to the ambient framing [4]. However, really, this set-up leads to a rather refined perspective on tangles as it provides a combinatorial framework for studying neighborhoods of ‘higher critical points’ (i.e. the points where tangles ‘change direction’). There is a tantalizing but mysterious connection of these critical points with classical differential ADE singularities [2]: on one hand, classical singularities seem to resurface as ‘perturbation-stable’ singularities in tangle diagrams, on the other hand, the differential machinery breaks down (producing ‘moduli of singularities’) in high parameter ranges and this simply cannot happen in the combinatorial approach; put differently, the combinatorial approach must be better behaved than the differential approach in some way. Certainly, the ‘higher compositional’ perspective given through the lens of diagrams is something that also has no differential counterpart at all (and it leads to new interesting observations, for instance, how to break up the classical three-fold symmetry of $D_4$ into a bunch of binarily-paired-up singularities, as we visualized in Figure 6). But despite many ‘visible patterns’, most of this line of research remains completely unexplored (attempts of laying at least some foundations were made in [4])… but maybe that’s what I find so exciting about it. :-)

Re #66: Lurie’s proof is sketched only for the topological case, of course.

I think a similar version can be developed for the geometric case, but this has yet to be done.

]]>Re #65, thanks. So nothing new when singularities meet geometric structure.

]]>Re #63: This is already considered in Lurie’s paper for Baas–Sullivan singularities, and a proof is sketched there.

]]>I have re-written (here) the Idea-section from scratch and expanded a fair bit – at least compared to what we had before. Added also more on the idea of the geometric cobordism hypothesis. Still, it all remains sketchy and could be expanded on much more.

]]>Does the geometric cobordism hypothesis approach handle the presence of singularities, so where it’s not just free generation from a point but from a set of generators?

]]>Looking through the page history, I see that it was me who has been shifting around that paragraph on the GTMW theorem in 2014. I forget what I was thinking, you are right that it doesn’t fit well where it is now and is hardly a motivation, as stated.

]]>If ever someone has the know-how and time, the page could do with some loving attention. A pleasanter opening Idea section would be a start.

Is that very obvious how that result in the third paragraph starting

As motivation, notice that…

is related to the preceding lines?

]]>added pointer to

- Dmitri Pavlov,
*The geometric cobordism hypothesis*, talk at*QFT and Cobordism*, CQTS (Mar 2023) [pdf, web]

Surely it would be better to have on this page Grady and Pavlov’s complete proof, rather than Lurie’s sketched proof.

]]>Clarified that Lurie’s paper gives a sketch and not a complete proof. Removed an unsubstantiated claim about a simplified proof to appear: the reference should be added once a proof is provided.

]]>Starting from Monday, July 4 at 9 am Central Daylight Time (UTC-5), Dan Grady and I will give a series of 4 lectures (90 minutes each) on the geometric cobordism hypothesis. BigBlueButton credentials can be obtained at https://www.carqueville.net/nils/GCH.html.

That’s great.

(When is the time to switch from saying “XYZ Cobordism Hypothesis” to “XYZ Cobordism Theorem”? Maybe a point could be made here.)

]]>Starting from Monday, July 4 at 9 am Central Daylight Time (UTC-5), Dan Grady and I will give a series of 4 lectures (90 minutes each) on the geometric cobordism hypothesis. BigBlueButton credentials can be obtained at https://www.carqueville.net/nils/GCH.html.

]]>Just to say that the postdoc job advertisements for our new “Center for Topological and Quantum Systems” (as per comment #53) is now out, see:

]]>I guess the citation (I think?) to “Topological and Quantum Systems” is to a document about the Center?

]]>re #34:

Just to say that our new **Center for Quantum and Topological Systems** is now live: here.

Part of granted activity is related to the intersection of

a) quantum programming languages

b) linear & modal homotopy type theory

c) twisted generalized cohomology theory

as indicated in the refined trilogy diagram that I am showing here. [edit: now I see the typo, will fix…]

We’ll be hiring a fair number of postdocs fairly soon. I’ll post the job openings as they become public.

]]>the necessity of incorporating the site Cart in the picture also led to consider the site FEmb_d as a natural extension. If we stayed in the purely topological case, then we could have tried to use spaces with an action of O(d) and may not have noticed the site of d-manifolds and open embeddings, which is more convenient to use in practice.

Thanks, that’s really interesting. I’ll try to find time to look at your locality article in more detail (when I got that darn proof typed up that’s absorbing my time and energy… ;-).

]]>So is it right that you are saying the geometric framed case is actually more amenable to direct proof than the topological case, but then implies it?

I would say that the geometric case inspired ways of thinking that we may not have encountered otherwise. For example, the necessity of incorporating the site Cart in the picture also led to consider the site FEmb_d as a natural extension. If we stayed in the purely topological case, then we could have tried to use spaces with an action of O(d) and may not have noticed the site of d-manifolds and open embeddings, which is more convenient to use in practice.

I have a vague memory that the remaining gap in the existing proof had to do with showing (or showing convincingly) that some space of Morse functions/handle decompositions is contractible or something like this. Is this an issue you solve or circumvent?

This was resolved by Eliashberg and Mishachev in 2011, who proved that the space of framed generalized Morse functions is contractible: https://arxiv.org/abs/1108.1000. This replaces Section 3.5 in Lurie’s paper.

We also develop a tool with similar functionality, in our locality paper this is Section 6.6 (the 1-truncatedness of our bordism categories corresponds to the contractibility of the space of framed generalized Morse functions used by Lurie to cut bordisms).

However, it is not quite accurate to say that this is “the remaining gap”, since some of the more important parts of Lurie’s argument in the other sections (3.1–3.4), such as Claim 3.4.12 and Claim 3.4.17, have their proofs omitted altogether, not even a sketch is present.

]]>Thanks. So is it right that you are saying the geometric framed case is actually more amenable to direct proof than the topological case, but then implies it?

I have a vague memory that the remaining gap in the existing proof had to do with showing (or showing convincingly) that some space of Morse functions/handle decompositions is contractible or something like this. Is this an issue you solve or circumvent?

]]>Re #48:

The locality paper is inseparable from the geometric cobordism hypothesis paper. So it’s not a “few pages”, but 40+41=81 pages (using version 2 of the GCH paper that will be uploaded soon).

For comparison, the sketch of a proof in Lurie’s paper is in Section 3.1 and 3.4 (and some fragments from 3.2, 3.3), which occupy 5+9=14 pages, plus a couple more for the relevant parts of 3.2, 3.3.

Some insights:

The locality property is invoked in the very first step of the proof to reduce to the geometric framed case. (In Lurie’s paper, Remark 2.4.20 instead deduces the locality principle from the cobordism hypothesis. However, very roughly this step corresponds in purpose to Section 3.2 there, which reduces to the case of unoriented manifolds instead of framed, although the actual details are completely different.)

The geometric framed case produces a d-truncated bordism category (before adding thin homotopies) because there are no nontrivial structure-preserving diffeomorphisms of d-dimensional bordisms embedded (or immersed) into R^d. This is used in the proof many times to simplify the arguments. (As far as I can see, there are no analogues in Lurie’s paper.)

The site FEmb_d (which encodes the geometric structures) plays a crucial role. In particular, encoding the homotopical action of O(d) using the site of d-manifolds and open embeddings is crucial for simplifying our proofs. (As far as I can see, there are no analogues in Lurie’s paper.)

Invariance under thin homotopies is encoded using a further localization of simplicial presheaves on FEmb_d. This is new and important for our proofs. In the topological case, this recovers precisely the (∞,d)-category of bordisms of Hopkins–Lurie, as opposed to just the d-category of bordisms. (As far as I can see, there are no analogues in Lurie’s paper.)

The machinery of the locality paper is used to establish the filtrations and pushout squares for the geometric framed case. (Very roughly, corresponds in purpose to Section 3.3 in Lurie’s paper (only the small part that is actually used in 3.4) and Claims 3.4.12 and 3.4.17, which are stated without proof (and without a sketch of a proof). The actual details are completely different, though.)

As an additional remark, all of these insights also apply to the topological cobordism hypothesis, even if we are not interested in the geometric case.

And even with these insights, the proof takes more than 80 pages.

]]>I admit not to have read the new article beyond the introduction, but without going into details, can one say in a few words what the key new insight is that makes the new proof happen?

Given that a fair bit of high-powered effort by several people had previously been invested into a proof of just the topological case while still leaving gaps, it’s a striking claim that not only this but also a grand generalization now drops out on a few pages. What is the new insight which makes this work and that previous authors had missed?

]]>We should add something about Daniel and Dimitri’s paper to the section For cobordisms with geometric structure.

Out of interest, is there a form of generalized tangle hypothesis of which this is the stabilization?

I recall we spoke about such things back at the n-Café here, following a conversation on $n$-categories of tangles as kinds of fundamental $n$-category with duals of stratified spaces. But the question was how to deal with geometric structure not just on the normal bundle.

]]>Yes, I think the topic is very much open. Mostly because essentially nobody seems to be attacking along that path.

That’s maybe not too surprising, since people tend to work on small steps for which there are existing hints of success, instead of embarking on one long journey through deserts and over mountains, from which one has not yet seen anyone come back.

And this is certainly somewhat puzzling about the heads-on approaches to non-perturbative Yang-Mills via AQFT or FQFT: that there are next to no hints for that or how it will work, when it works. There is no partial result and few heuristic arguments for what it is that will eventually make the zoo of hadron masses come out by these approaches. If and when it eventually works, it is going to be a dramatic success of pure abstract thinking.

That’s why I came to feel that the alternative approach via holographic QCD inside M-theory is more promising: even though it superficially *sounds* more crazy, there is a wealth of hints for *how* it makes things work (all those strings are, after all, the original and still the best idea for how confinement works, namely via tensionful color flux tubes) and, more importantly, hints *that* it works (from the close match of the zoo of predictions/measurements of hadron masses here).

$\,$

By the way, it should be only the moduli stack of fields which is a stack of non-abelian differential cocycles. But the higher pre-quantum gerbe on that stack should be abelian, and the quantization should be by push-forward in some abelian cohomology twisted by that pre-quantum gerbe.

This is, in any case, what happens in familiar examples, notably this is how the quantization of 3d Chern-Simons theory works (*geometric quantization by push-forward*) where the push-forward is in differental K-theory but computes the quantum states for non-abelian CS theory.

It is this picture of quantization via push-forward in abelian (“linear”) cohomology over non-abelian stacks which I was after in *Quantization via Linear homotopy types*.

Re #44:

I see, so would it be correct to say that although we may expect the classical Yang–Mills theory to involve gerbes and similar stuff, this has not yet been figured out in details?

Your project on Lepage gerbes looks incredibly interesting.

Here is my line of thought on constructing quantized FFTs:

Produce the prequantum data (some form of nonabelian differential cohomology) from the classical data.

Convert the prequantum data in (1) to a map like in the right side of the geometric cobordism hypothesis.

Use the geometric cobordism hypothesis to convert (2) to a fully extended functorial field theory.

Integrate the prequantum data in (1) using pushforwards in nonabelian differential cohomology, producing another map like on the right side of the GCH.

Use the GCH to convert (4) to a fully extended functorial field theory, which is the quantization of (3).

Status so far:

(3) and (5) are supplied by the geometric cobordism hypothesis.

(2) is current work in progress (the third paper in the series), and should be out soon.

(4) is planned (the fourth paper in the series), in principle we know what to do.

(1) was missing so far, but it seems like your work with Khavkine provides a complete solution.