### Hopkins-Lurie on Baez-Dolan

#### Posted by Urs Schreiber

I just heard at HIM from Mike Hopkins apparently the kind of talk that Jacob Lurie gave a while ago, as recounted here.

It’s about their work on formulating the *Baez-Dolan tangle hypothesis*/*extended TQFT hypothesis* (which essentially says that every extended TQFT is already fixed by the “$n$-space of objects” which it assigns to the point) within $\infty \;n$-categories and then proving it (proof done for low $n$ and sketched for all $n$).

The key ingredients (as presented in the talk) are these:

1) We need this definition: an $\infty$ 0-category is a topological space or something Quillen equivalent to it (a Kan complex aka $\infty$-groupoid, for instance). An $\infty\; n$-category is recursively defined to be a category enriched over $\infty\; (n-1)$-categories (enriched in the ordinary strict sense!).

then the point is: for all $n \in N$ framed $n$-manifolds naturally form an $\infty\; n$-category $\mathrm{Bord}_n^{fr}$ whose $(k \leq n)$-morphisms are framed cobordisms and whose $(k \gt n)$-morphisms are diffeomorphisms and higher homotopies between these.

A framed cobordisms here is a $(d \leq n)$-dimensional cobordism $\Sigma$ equipped with a trivialization of the $n$-stabilized tangent bundle $T \Sigma \oplus (\Sigma \times\mathbb{R}^{n-d}) \stackrel{\simeq}{\to} \Sigma \times \mathbb{R}^n$.

2) The notion of *very dualizable* objects (Jacob Lurie’s idea): to me this is the notion of $\infty$-equivalence with “invertible up to” everywhere replaced by “adjoint up to”.

recursive Def.:

an *$\infty\; 2$-category with adjoints* is one in which each 1-morphism has a left and right adjoint.

an *$\infty\; n$*-category with adjoints is an $\infty \;n$-category such that all Hom-($\infty\; (n-1)$-categories) are $\infty (n-1)$-categories with adjoints.

For every $\infty \;n$-category
$C$
write
$C^f$ for the largest $\infty \;n$-category with adjoints sitting inside. The “f” is for *finite* since having lots of adjoints is imposing lots of conditions which usually say that things are finite dimensional (simplest example: a vector space which has a dual such that it is the dual of its dual is finite dimensional).

This finiteness condition here ultimately is supposed to relate to things such as the “rational” in rational CFT (which says that certain 2-vector spaces are finite dimensional, i.e. that certain monoidal categories have finitely many simple objects).

Now finally: an object $V \in C$ for $C$ symmetric monoidal is called *very dualizable* if it is dualizable in $C^f$. That means that unit and counit 1-morphisms of the duality have to have 2-sideed adjoints whose structure 2-morpshisms have 2-sided adjoints, and so on.

Write $C^{fd}$ for the “space” of all very dualizable objects in the symmetric monoidal $\infty \; n$-category $C$. This turns out to be always an $\infty\; 0$-category, i.e. “really a space”.

(I asked afterwards if he knew about Eugenia Cheng’s result that An $\omega$-category with all duals is an $\omega$-groupoid, but he didn’t.)

Then we have their

**Theorem (Baez-Dolan hypothesis).** *The space (i.e. $\infty\; 0$-category) of $\infty\; n$-functors
$\mathrm{Bord}_n^{fr} \to C$
is the space (i.e. $\infty\; 0$-category) of very dualizale objects in $C$:
$hom(Bord_n^{fr}, C) \simeq C^{fd}
\,.$
*

Mike Hopkins says they have a detailed proof for $n \leq 2$ and a sketch for arbitrary $n$. He pointed out that Bartels, Douglas and Henriques have a similar statement for *strict* $n$-categories going up to $n \leq 7$.

This is in particular equivalent to saying that (hope I get this right now): $\mathrm{Bord}_n^{fr}$ is the free $\infty \; n$-category on a single *very dualizable* object.

Mike Hopkins mentioned a curious direct **corollary** of this:

there is an action of $\mathrm{GL}_n(\mathbb{R})$ on $\mathrm{Bord}_n^{fr}$ which rotates the framing everywhere. By the above theorem this now means there is a $GL_n(\mathbb{R})$-action on the space of of very dualizable objects.

Then came something which I now see I still don’t fully understand. He writes $\mathbf{B} G$ for the 1-object groupoid version of the group $G$ (well, the boldface is the notation we are using here on the $n$-Café, so I am happy with that) then uses $SO(2) \simeq B \mathbb{Z}$ to say that there is a functor from $\mathbf{B} \mathbb{Z}$ to the space of very dualizable objects, which means that the above $GL_n(\mathbb{R})$-action in particular picks one automorphism of every very dualizable object. In fact, that can be built easily from using the structure maps of the duality on it.

Sorry, I’ll straighten that last part out and get back to you then.

## Re: Hopkins-Lurie on Baez-Dolan

This is fantastic! I saw Hopkins give a talk a while a go, it was sort of a dumbed down version of the first distinguished lecture talk he gave at the fields institute (which is a available in streaming audio) I am gonna have to go over this a lot more. I think this stuff is really pretty amazing!

thanks for posting on it.

sean