The Thinker Musings

According to Atiyah, a $d$-dimensional TQFT is a tensor functor, $Z$, from the category, $Cob(d)$, whose objects are closed $(d-1)$-manifolds¹ , and whose morphisms are bordisms $$Hom (M,N) = \{B: \partial B \simeq M\amalg \overline{N}\}/\text{diffeomorphisms}$$ to $\Vect$, the category of complex vector spaces. $Cob(d)$ is a symmetric monoidal category, with $\otimes$ given by disjoint union of manifolds, and $Z$ preserves tensor products $$\begin{gathered} Z(M\amalg N) \simeq Z(M)\otimes Z(N)\\ Z(\emptyset) \simeq \mathbb{C} \end{gathered}$$

The vague idea of extended TQFT is to replace $Cob(d)$ with some sort of $n$-category (where $n=d$), consisting of manifolds of all dimension $m\leq d$, and replace $Vect$ with a similarly fancied-up $n$-category, $\mathcal{C}$.

Over the course of several lectures, Jacob proposes to tell us exactly what these all are, but the vague version is as follows:

For $m\leq d$, a $d$-framing of a manifold, $M$, of dimension $m$, is a trivialization of $T_M\oplus \mathbb{R}^{d-m}$.

$Cob(d)^{\text{framed}}_{\text{ext}}$ is a symmetric monoidal $d$-category (with tensor product given by disjoint union of manifolds).

• Objects are $0$-manifolds with a $d$-framing.
• Morphisms are $d$-framed bordisms between $d$-framed $0$-manifolds.
• 2-Morphisms are $d$-framed bordisms between $d$-framed $1$-manifolds.
• ⋮
• $d$-morphisms are $d$-framed $d$-manifolds (with corners) $/\text{diff}$

The statements which he proposes to prove are that

Given a $d$-category, $\mathcal{C}$, with tensor product, an ETQFT “with values in $\mathcal{C}$” is a tensor functor

1. $Z:,円 Cob{d}^{\text{framed}}_{\text{ext}}(d) \to \mathcal{C}$
2. The “fully dualizable” objects, $X\in \mathcal{C}$ are given by $X = Z(\bullet)$.

In some sense, the whole ETQFT is determined by knowing what $Z$ of a point is. Here, “fully dualizable objects” is some condition analogous to demanding that the vectors spaces in an ordinary TQFT, associated to a $(d-1)$ manifold, are finite-dimensional.

There is, of course, a 110 page paper providing “an outline” of the idea. Probably, it will be more intelligible than my summary.

(some might say a precise statement of) the Baez-Dolan Cobordism Hypothesis

My impression is that nobody would claim that the original statement was already technically precise, nor that it was suggested to be. Part of the aspect of proving it is to find the natural formalization that makes it naturally true. In this respect the tangle hypothesis is not unlike the homotopy hypothesis, I’d think.

Re: formalizing it

I’d like to understand the import of the framing in this story. Most manifolds aren’t framable after all (referring to the top dimension where we don’t get any extra stuff to play around with). Later on in the notes, Lurie gives a version where the framing is removed and reasonably arbitrary other structures are substituted. So, I guess I’m wondering if the framing is just a math trick or is something deeper. Of course, I’m only on pg 50 of the notes, so maybe this is addressed later.

Re: New beginnings

hi,
i got on a search quest today and found your site.
I suppose you heard of the “1982 aspect experiment” ?
Can you tell me what it actually was… the experiment itself?

more specifically, what does math and the experiment have to do with each other?

People say its proves this and that…
other say no.
or it brings up more questions than it answers.

anyway, thanks for a reply :)