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added at cobordism hypothesis a pointer to
where the case for $(\infty,1)$-categories is spelled out and proven in detail.
At cobordism hypothesis the section titled For noncompact cobordisms used to contain nothing but a link to Calabi-Yau object. I have now added a few lines of text at least, trying to convey the rough idea.
added also a section For cobordisms with singuarities (boundaries/branes and defects/domain walls) with just a few lines on the idea for the moment, just enough to highlight theorem 4.3.11.
(Which is easily missed; much of the best magic happens on the last 10 pages of 111…)
What would be the established term for these “diagrams indicating types of singularities” on which the cobordism-with-singularities hypothesis/theorem says that the cobordism-with-singularities $(\infty,n)$-category is freely generated from as a symmetric monoidal $(\infty,n)$-category with all duals?
So I mean for instance the simple diagram
$0 \longrightarrow \ast$indicating a domain wall separating the left phase (“0”) from the right (“\ast”). The archetypical path in a fundamental category crossing a stratum.
May these be called “catastrophe diagrams”? Is there any half-way established term available?
So paths which go up and down through the strata, like we discussed once? I wonder if there’s a term in a paper like Diagrammatics, Singularities, and their Algebraic Interpretations.
Taking a look at the workshop you’re attending, I see Catherine Meusburger is speaking on ’Diagrams for Gray categories with duals’, which is based on Gray categories with duals and their diagrams. Todd gets a mention:
The definition of a diagrammatic calculus for Gray categories follows the pattern for categories and 2-categories. The diagrams are a three-dimensional generalisation of the two-dimensional diagrams defined above, and were previously studied informally by Trimble [31].
added to cobordism hypothesis in the section on the framed version a brief paragraph Implications – The canonical O(n)-action on fully dualizable objects
(this statement used to be referred to further below in the entry, but wasn’t actually stated)
Added corresponding cross-pointers to dual object ($n = 1$) and to orthogonal spectrum ($n = \infty$). Somebody needs to create an entry for Serre automorphism ($n =2$).
How does that result fit with the $(\mathbb{Z}_2)^n$ automorphism $\infty$-group of $(\infty, n)$-Cat? What are the fully dualizable objects of the latter?
On the other hand, what would happen if we opted instead for a profunctor or span-like approach? Objects there are generally fully dualizable?
I see just out there is Rune Haugseng’s Iterated spans and “classical” topological field theories. Urs and Joost get cited.
In §6 we then prove that $Span_n(C)$ is symmetric monoidal and that all its objects are fully dualizable
Oh, but
Conjecture 1.3 (Lurie). The $O(k)$-action on the underlying ∞-groupoid of $Span_k(C)$ is trivial, for all ∞-categories $C$ with finite limits.
Anyway, I’m more interested with my first question.
Christopher Schommer-Pries has some useful notes – Dualizability in Low-Dimensional Higher Category Theory.
I guess it’ll only be the terminal object which is dualizable.
Thanks for highlighting Haugseng’s article! Would have missed that otherwise.
Have added pointers to his results to (infinity,n)-category of correspondences and elsewhere.
added a tad more on the definition of cobordisms with (X,xi)-structure and then added in particular the proof idea of how the cobordism hypothesis for $(X,\xi)$-structure follows from the framed case
added now also the proof of 2.4.26 from the $(X,\xi)$-version, i.e. the reduction to the special case that $X = B G$. This is of course just a straightforward corollary, but I have added a line discussing how the result is really the correct concept of homotopy invariants in the sense defined/discussed at infinity-action.
added now also the proof of 2.4.26 from the $(X,\xi)$-version, i.e. the reduction to the special case that $X = B G$. This is of course just a straightforward corollary, but I have added a line discussing how the result is really the correct concept of homotopy invariants in the sense defined/discussed at infinity-action.
What’s the status of the generalized tangle hypothesis?
This appears as theorem 4.4.4 in Lurie’s writeup.
Started a section on the case of (un-)oriented field theories.
After recalling some of the statements from Lurie’s article, I am after making explicit the following corollary. While being a simple corollary, this way of stating it explicitly is immensely useful for the study of unoriented local prequantum field theories. I wonder if this has been made explicit “in print” elsewhere before:
Let $Phases^\otimes \in Ab_\infty(\mathbf{H})$ be an abelian ∞-group object, regarded as a (∞,n)-category with duals internal to $\mathbf{H}$.
At least if $\mathbf{H} =$ ∞Grpd, then local unoriented-topological field theories of the form
$Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$are equivalent to a choice
of $X \in \mathbf{H}$ equipped with an $O(n)$-∞-action
a homomorphism of $O(n)$-∞-actions $L \colon X \to Phases$ (where $Phases^\otimes$ is equipped with the canonical $\infty$-action induced from the framed cobordism hypothesis), hence to morphisms
started disucssion of some simple but interesting examples at local prequantum field theory – Higher CS theory – Levels.
But I am being interrupted now…
I followed up that proposition Exchanging fields for structure with a remark amplifying its relevance/meaning.
Given a “structure”, i.e. an $(X,\zeta)$-structure in the terminology of Lurie’s writeup, and hence given $Bord_n^{(X,\zeta)}$, what is actually a direct way (i.e. not via the full cobordism hypothesis) to define the “$(X,\zeta)$-diffemorphism group” of an $n$-dimensional manifold $\Sigma$, i.e.
$\Pi(Diff_{(X,\zeta)}(\Sigma)) \coloneqq \Omega^n_{\Sigma} Bord_n^{(X,\zeta)}$?
(Notice: no geometric realization on the right.)
I think I know what it is, but I am a little vague on how to formally derive this from the “definition” of $Bord_n^{(X,\zeta)}$.
I think the right answer is to form the homotopy pullback along the canonical map
$Diff(\Sigma) \longrightarrow \mathbf{Aut}_{/BO(n)}(\Sigma)$of automorphisms in the “slice of the slice” over the classifying map $X \to BO(n)$ of $\zeta$.
I have spelled this out now as def. 3.2.9 on page 34 at Local prequantum field theory (schreiber). After that definition there are spelled out proofs that with this defintion we do get the expected higher extensions of $B Diff(\Sigma)$.
So this looks right. But if anyone cares to give me a sanity check, that would be appreciated.
I have forwarded that question to MO
Shouldn’t the statement of the cobordism theorem either assume that $\mathcal{C}$ has duals, or alternatively have the map $\mathrm{pt}^*$ land in the core of $\mathcal{C}^\mathrm{fd}$ rather than in the core of $\mathcal{C}$ itself?
(I would have fixed this myself if I was absolutely sure about it.)
Woops. Yes, you are absolutely right. I have added the qualifier “with duals” at the beginning of the Statement.
But the entry would deserve some further polishing. If you feel energetic about this at the moment, you should edit it.
Added the reference
This claims to have the proof modulo conjecture 1.2, which is to appear shortly. Has it appeared?
Is there as yet a published complete proof of the cobordism hypothesis. The page says
This is almost complete, except for one step that is not discussed in detail. But a new (unpublished) result by Søren Galatius bridges that step in particular and drastically simplifies the whole proof in general.
Do we know if the status has changed?
Yes, it’s time this was sorted. Lurie having proved it but then not supplying all details I’m sure killed off other people’s incentive to work on it, since the credit had already been claimed.
Last I checked, the proof of that conjecture by Ayala-Francis has not appeared. That was a few months back and I should check again. But clearly there is no evident public announcement of the proof.
On the culture of announcing conjectures in homotopy theory as theorems, see also Clark Barwicks’s The future of homotopy theory (pdf)
The politically efficient way to proceed is shown by number theory: Huge excitement built up by conjectures, even if their actual content doesn’t mean much to most researchers.
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