The n-Category Café

I think this stuff is really pretty amazing!

What I find amazing about it – but that already pertains to the original Baez-Dolan hypothesis – that this statement is so fundamental on the mathematical side.

One might have imagined that quantum field theory turns out to be an intrinsically intricate and complicated concept. But this theorem tells us, at least for TQFT, that it is one of the most “tautological” structures in math. As we discussed elsewhere, by some magic the free stable $\infty$-groupoid on a single dualizable object is just another incarnation of the sphere spectrum. It doesn’t get much more “deep” and “foundational” in math than the sphere spectrum, I suppose. If you know what I mean.

So the Baez-Dolan hypothesis is an amazing intersection of abstract nonsense with deep pure-math structure and with fundamental physics. Quite amazing.

Oops. The $U$ there was supposed to be an $O$.

The point was that in $GL(n,\mathbb{R})$ we have in particular sitting $SO(2) \simeq U(1)$. But I got confused about the supposed consequences of this statement anyway. Still am.

Eugenia,

thanks for asking. Let me check this again. I’ll try to get back to you on this tomorrow.

Impressive!

Like the cobordism hypothesis, the tangle hypothesis can be generalized in many ways. For example, one can consider embedded submanifolds with tangential structure more complicated than that of a $k$-framing and embedded submanifolds with singularities. In these cases, one can still use the argument sketched above to establish a universal property of the relevant $(\infty, n)$-category.

It’s time to break out the champagne!

Here is further thought. I’ll expand on my remarks above on the multi-(co)-span description of cobordisms and QFTs above in that

a) I talk about the gluing of common boundaries of a single cobordism which I mentioned above in more detail;

b) describe how it maps to the groupoidified trace operation (it’s obvious, this must have been discussed explicitly somewhere?);

and

c) prove the Willerton-Bartlett crossing-with-the-circle equation in this context, which asserts that an extended TQFT $Z$ assigns to a cobordism of the form $\Sigma \times S^1$ the endomorphisms on the identity on what it assigns to $\Sigma$ itself:

$$Z(\Sigma \times S^1) \simeq End_{Id_{Z(\Sigma)}} \,.$$

Points a) and b) by way of concrete example:

Let $$\array{ && I \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt }$$

be the standard co-span version of the interval $I = [0,1]$. To be thought of as a “straight” interval.

The “co-trace” over this is obtained by

- first dualizing on the output to arrive at the co-span

$$\array{ && I \\ & {}^{in \sqcup out}\nearrow && \nwarrow \\ pt \sqcup pt &&&& \emptyset } \,,$$

to be thought of as a curved/bended interval, and then composing on the left with the result of taking the identity

$$\array{ && pt \\ & {}^{Id}\nearrow && \nwarrow^{Id} \\ pt &&&& pt }$$

and similarly dualizing the input to an output to arrive at

$$\array{ && pt \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} \\ \emptyset &&&& pt \sqcup pt }$$

The compsite span is the pushout

$$\array{ &&&& S^1 \\ &&& \nearrow && \nwarrow \\ && pt &&&& I \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} && {}^{in \sqcup out}\nearrow && \nwarrow \\ \emptyset &&&& pt \sqcup pt &&&& \emptyset }$$ namely the circle. The co-trace over the interval cobordism.

Dually, if we have a span

$$\array{ && P \\ & {}^{x}\swarrow && \searrow^{y} \\ X &&&& X }$$

describing a “geometric function” or “groupoidified linear map” or whatever we’ll eventually agree on calling it, in that for instance $X$ is a finite set with $n$ elements and $P \to X \times X$ is a bundle of sets, i.e. a set-valued $n \times n$ matrix, thought of as a categorified $\mathbb{N}$-valued $n\times n$-matrix, then the co-version of the above procedure produces the trace of $P$

$$\array{ &&&& tr(P) \\ &&& \swarrow && \searrow \\ && X &&&& P \\ & {}^{}\swarrow && \searrow^{Id \times Id} && {}^{x \times y}\swarrow && \searrow \\ pt &&&& X \times X &&&& pt }$$

in that $tr(P)$ is, in the simple example mentioned, the set $tr(P) = \sqcup_{x \in X} P_{x,x}$.

So, now consider a $\sigma$-model QFT given by a background field which is given by an $n$-bundle $P \to X$ over some target space $X$. By general argument mentioned before, this is the multi-span map given by homming termwise into $P$

$$QFT_P := [-, P]$$

where the resulting spans, for instance

$$QFT_P\left( \array{ && \Sigma \\ & {}^{in}\nearrow && \nwarrow^{out} \\ \Sigma_{in} &&&& \Sigma_{out} } \right) = \array{ && [\Sigma, P] \\ & {}^{in^*}\swarrow && \searrow^{out^*} \\ [\Sigma_{in},P] &&&& [\Sigma_{out},P] }$$

is regarded as a groupoidfied linear map which acts on generalized sections $\Psi \in \Gamma([\Sigma_{in}, P])$ which are themselves spans

$$\array{ && \Psi \\ & \swarrow && \searrow \\ \Omega_{pt}F &&&& [\Sigma_{in},P] } \,,$$

(where $\Omega_{pt}F$ is the “ground monoid”, i.e. the endomorphism monoid of the point in the fiber $F$ of $P$ – but that’s not relevant for the argument right now)

by composition of spans.

So we can check what our $QFT_P$ does to cobordiusms of the form $\Sigma \times S^1$ by evaluating this prescription for that case.

For ease of notation assume $\Sigma$ has no in- and output. So that $\Sigma \times I$ is given by a co-span

$$\array{ && \Sigma \times I \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt }$$

So the point then is that we know what $QFT_P$ does to this co-span and, since it respects composition in multi-spans (from co-multispans to multispans) we know that to $\Sigma \times S^1$ the $QFT_P$ assigns the trace

$$\array{ &&&& tr[\Sigma \times I, P] \\ &&& \swarrow && \searrow \\ && [\Sigma, P ] &&&& [\Sigma \times I, P] \\ & {}^{}\swarrow && \searrow^{Id \times Id} && {}^{d_0 \times d_1}\swarrow && \searrow \\ pt &&&& [\Sigma,P] \times [\Sigma, P] &&&& pt }\,.$$

Using the hom-adjunction this is the same as the pullback

$$\array{ &&&& tr[\Sigma \times I, P] \\ &&& \swarrow && \searrow \\ && [\Sigma, P ] &&&& [I,[\Sigma , P]] \\ & {}^{}\swarrow && \searrow^{Id \times Id} && {}^{d_0 \times d_1}\swarrow && \searrow \\ pt &&&& [\Sigma,P] \times [\Sigma, P] &&&& pt }\,.$$

Now, maps into $[I,[\Sigma,P]]$ are homotopies (morphisms) between two maps into $[\Sigma,P]$. The pullback diagram says that maps into $tr[\Sigma \times I, P]$ are homotopies (morphisms) between one maps into $[\Sigma,P]$ and itself.

This in turn says that regarded as the corresponding groupoidied space of sections, the value $QFT_P(\Sigma \times I)$ is the collection of spans

$$\array{ && \Psi \\ & \swarrow && \searrow \\ \Omega_{pt}F &&&& tr([\Sigma \times I,P]) } \,.$$

By the universal pullback property of $tr([\Sigma \times I,P])$ just mentioned every such span is indeed an endomorphism of a section of $[\Sigma,P]$.

To get the full Willerton-Bartlett statement play with the hom-adjunciton a bit more…

A further remark on span trace:

it would be good to see how the “span trace” for spans of categories relates to the notion of trace on an endofunctor described by Ganter-Kapranov # and by Bartlett-Willerton #.

So let $C$ be a category and $F : C \to C$ an endofunctor. Bartlett-Ganter-Kapranov-Willerton (alphabetical! I am trying to attribute this correctly this time…) say that the categorical trace of $F$ is the collection of transformations $Id_C \Rightarrow F$.

On the other hand, $F$ defines the span

$$\array{ && C \\ & {}^{Id}\swarrow && \searrow^{F} \\ C &&&& C }$$

and by the general nonsense of span trace the trace of this span is the pullback over

$$\array{ C &&&& C \\ & {}_{Id \times Id} \searrow && \swarrow_{Id \times F} \\ && C \times C } \,.$$

One should take the suitable pseudo or lax pullback. What is it? (I am asking the same question here).

I am not fully sure yet, but let me approximate this question a bit.

Since I want to compute something like a homotopy limit, I should maybe replace my diagonal $C \stackrel{Id \times Id}{\to}C \times C$ by its factorization

$C^I \stackrel{d_0 \times d_1}{\to}C \times C$

for $I = \{a \to b\}$ the interval category. (Thinking of this as coming from homming out of the co-span co-trace this corresponds to making duality on objects not an operation at a point but smeared over an interval).

So let me for the moment consider the ordinary (1-categorical) limit over

$$\array{ C^I &&&& C \\ & {}_{d_0 \times d_1} \searrow && \swarrow_{Id \times F} \\ && C \times C } \,.$$

That limit, say $R$ for need of a symbol, should be the full subcategory of the arrow category $C^I$ on objects (morphisms in $C$) of the form $$a \to F(a) \,,$$ for $a \in Obj(C)$. I.e. the “lax fixed points of $F$”.

So how does that relate to Bartlett-Ganter-Kapranov-Willerton? Well, this $R$ is the object which represents their categorical trace.

Just for the record, I’ll collect the links here:

Distler on Lurie on Baez-Dolan, part I: fundamental $\infty$-groupoids.