### New beginnings

Today was my first lecture of the new semester, there was some sort of hubbub on The Mall in Washington, and Jacob Lurie gave the first of a series of lectures on “Extended” Topological Quantum Field Theory and a proof of (some might say a precise statement of) the Baez-Dolan Cobordism Hypothesis.

According to Atiyah, a $d$-dimensional TQFT is a tensor functor, $Z$, from the category, $Cob(d)$, whose objects are closed $(d-1)$-manifolds^{1}, 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

- $Z:\, Cob{d}^{\text{framed}}_{\text{ext}}(d) \to \mathcal{C}$
- 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.

^{1} Here, and below, a “manifold” is smooth, compact and oriented. Usually, it will be a manifold with boundary. When we want to denote a manifold without boundary, we’ll call it “closed.”

## formalizing it

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.