The n-Category Café

No — I’ve posed this as a challenge here before, but with no success. You can go through formulas involving $\pi$ and look for good ones…

The best I can do is this. Let the groupoid $X$ be the coproduct of all groups isomorphic to $\mathbb{Z}/n \times \mathbb{Z}/n$ for some $n$. Then

$$|X| = \frac{\pi^2}{6}$$

$\pi$ isn’t such a nice number in a lot of ways, and I personally would be neither surprised nor disappointed if no one could suggest a nice way for $\pi$ to be the cardinality of something. $e$, on the other hand, clearly deserves to be the cardinality of something natural.

My guess is that an object of $RSet_g$ is like a stick, and a morphism is a way of taking a stick, breaking it up into countably many shorter sticks, rearranging them sticks, and putting them back together to form another stick.

More precisely: I think there’s a skeleton of $RSet_g$ with one object for each interval $[0,L)$ with $0 \le L \le \infty$, where a morphism

$$f : [0,L) \to [0,L)$$

is a bijection that’s piecewise linear on countably many subintervals of the form $[a,b)$ whose disjoint union is $[0,L)$, with slope 1 on each of these subintervals.

I’m not sure this is true: in particular, there are some quite wild ways of decomposing $[0,L)$ as a disjoint union of countably many subintervals of the form $[a,b)$, and I’m not sure we should allow all of them. (If we take the right-hand endpoints of these subintervals, they may have cluster points, and the cluster points may have cluster points, and so on.)

Nonetheless I feel pretty sure of the basic picture, that positive real sets are like sticks of rigid material, which we can break apart and rearrange. It would be nice to prove a precise version of this, if only to describe $RSet_g$ in a way that non-category-theorists could understand.

It also looks, on a brief glance, that they only conjecture but do not prove that the functor $FinSet_g \to RSet_g$ is fully faithful?

I didn’t read everything in the paper carefully. According to my guess above, this would be true. The objects of $FinSet_g$ can be seen as sticks whose lengths are natural numbers, and a morphism is a way of breaking up such a stick into finitely many pieces of the same sort and rearranging the pieces.

According to my guess above, this would be true.

Really? Why couldn’t you break a stick of natural-number length into pieces of fractional length?

Hmm, okay, I guess I see.

Your description also seems to imply that the automorphism groups of every object of $RSet_g$ except $0$ and $\infty$ are isomorphic? (Send $f:[0,L) \to [0,L)$ to $(\lambda t. \frac{M}{L} f(\frac{L}{M} t)) : [0,M) \to [0,M)$.) In other words, $RSet_g \cong 1 + \sum_{0\lt L\lt \infty} \mathbf{B} G + \mathbf{B} G_\infty$, where $G$ and $G_\infty$ are two yet-to-be-determined groups.

True, people should call these entities something else unless and until they seem to act enough like “sets” to deserve the name.

It’s worth remembering that for a long time many mathematicians didn’t believe imaginary or even negative numbers were really “numbers”. As late as 1831, in his book On the Study and Difficulties of Mathematics, Augustus de Morgan wrote:

The imaginary expression ${\displaystyle {\sqrt {-a}}}$ and the negative expression $-b$ have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are equally imaginary, since $0 - a$ is as inconceivable as ${\displaystyle {\sqrt {-a}}}$.

By now most mathematicians are happy to call these things ‘numbers’. So, it’s possible that we’ll eventually update our concept of ‘set’ to include sets with arbitrary real cardinalities. However, they’d have to prove their usefulness first. For now it’s best to keep thinking about these issues and seeing what we can do.

He told me a bit about derived symplectic geometry, and how it permits all sorts of wonderful improvements to the theory of Lagrangian correspondences. I guess some relevant papers are:

• Bertrand Toen, Derived algebraic geometry and deformation quantization.

• T. Pantev, B. Toen, M. Vaquie and G. Vezzosi, Shifted symplectic structures.

and the review paper:

• T. Pantev and G. Vezzosi, Symplectic and Poisson derived geometry and deformation quantization

Then we talked about something I’m really fascinated by: symplectic geometry in thermodynamics. I’ll probably blog about this sometime (continuing this series).

I have no idea what those would be like. I’ll get back to you next century.

Nice question!

I think one can make sense of an $q$-category where $q$ is a positive fraction, just by mimicking the usual definition of a fraction, and using the Gray tensor product. I.e. define a $q$-category to be a pair $(A,B)$ of an $m$-category $A$ and an $n$-category $B$ for some integers $m$ and $n$, and then define an equivalence relation $(A,B) \sim (C,D)$, if $C$ is an $m'$-category and $D$ is an $n'$-category by asking that $A \otimes D \simeq B \otimes C$, where $\otimes$ is the Gray tensor product of $\infty$-categories (viewing all the higher categories involved as $\infty$-categories), and $\simeq$ is equivalence of $\infty$-categories. Here by $\infty$-categories I mean $(\infty, \infty)$-categories, of course.

I think this construction (to the extent to which it is possible to carry out at the present time) does actually categorify the usual construction of positive fractions from natural numbers, if we view a natural number $n$ as an $n$-category with a single object and a single non-identity $n$-arrow and all $m$-arrows for $m \lt n$ identities. I haven’t thought about it very carefully, though.

Whether this is good for anything, I have no idea! It’s quite fun to speculate upon though!