November 23, 2017

Real Sets

Posted by John Baez Good news! Janelidze and Street have tackled some puzzles that are perennial favorites here on the $n$-Café:

• George Janelidze and Ross Street, Real sets, Tbilisi Mathematical Journal, 10 (2017), 23–49.

Abstract. After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions:

• what is a set with half an element?

• what is a set with π elements?

The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories series monoidal and conclude by only briefly mentioning the non-commutative possibility called ω-monoidal. We include some remarks on sets having cardinalities in $[-\infty,\infty]$.

First they define a series magma, which is a set $A$ equipped with an element $0$ and a summation function

$\sum \colon A^{\mathbb{N}} \to A$

obeying a nice generalization of the law $a + 0 = 0 + a = a$. Then they define a series monoid in which this summation function obeys a version of the commutative law.

(Yeah, the terminology here seems a bit weird: their summation function already has associativity built in, so their ‘series magma’ is associative and their ‘series monoid’ is also commutative!)

The forgetful functor from series monoids to sets has a left adjoint, and as you’d expect, the free series monoid on the one-element set is $\mathbb{N} \cup \{\infty\}$. A more interesting series monoid is $[0,\infty]$, and one early goal of the paper is to recall Higgs’ categorical description of this. That’s Denis Higgs. Peter Higgs has a boson, but Denis Higgs has a nice theorem.

First, some preliminaries:

Countable products of series monoids coincide with countable coproducts, just as finite products of commutative monoids coincide with finite coproducts.

There is a tensor product of series monoids, which is very similar to the tensor product of commutative monoids —- or, to a lesser extent, the more familiar tensor product of abelian groups. Monoids with respect to this tensor product are called series rigs. For abstract nonsense reasons, because $\mathbb{N} \cup \{\infty\}$ is the free series monoid on one elements, it also becomes a series rig… with the usual multiplication and addition. (Well, more or less usual: if you’re not familiar with this stuff, a good exercise is to figure out what $0$ times $\infty$ must be.)

Now for the characterization of $[0,\infty]$. Given an endomorphism $f \colon A \to A$ of a series monoid $A$ you can define a new endomorphism $\overline{f} \colon A \to A$ by

$\overline{f} = f + f\circ f + f \circ f \circ f + \cdots$

where the infinite sum is defined using the series monoid structure on $A$. Following Higgs, Janelidze and Street define a Zeno morphism to be an endomorphism $h \maps A \to A$ such that

$\overline{h} = 1_A$

The reason for this name is that in $[0,\infty]$ we have

$1 = \frac{1}{2} + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 + \cdots$

putting us in mind of Zeno’s paradox:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal. — Aristotle, Physics VI:9, 239b10.

So, it makes lots of sense to think of any Zeno morphism $h \colon A \to A$ as a ‘halving’ operation. Hence the name $h$.

In particular, one can show any Zeno morphism obeys

$h + h = 1_A$

Higgs called a series monoid equipped with a Zeno morphism a magnitude module, and he showed that the free magnitude module on one element is $[0,\infty]$. By the same flavor of abstract nonsense as before, this implies that $[0,\infty]$ is a series rig…. with the usual addition and multiplication.

Categorification

Next, Janelidze and Street categorify the entire discussion so far! They define a ‘series monoidal category’ to be a category $A$ with an object $0 \in A$ and summation functor

$\sum \colon A^{\mathbb{N}} \to A$

obeying some reasonable properties… up to natural isomorphisms that themselves obey some reasonable properties. So, it’s a category where we can add infinite sequences of objects. For example, every series monoid gives a series monoidal category with only identity morphisms. The maps between series monoidal categories are called ‘series monoidal functors’.

They define a ‘Zeno functor’ to be a series monoidal functor $h \colon A \to A$ obeying a categorified version of the definition of Zeno morphism. A series monoidal category with a Zeno functor is called a ‘magnitude category’.

As you’d guess, there are also ‘magnitude functors’ and ‘magnitude natural transformations’, giving a 2-category $MgnCat$. There’s a forgetful 2-functor

$U \colon MgnCat \to Cat$

and it has a left adjoint (or, as Janelidze and Street say, a left ‘biadjoint’)

$F \colon Cat \to MgnCat$

Applying $F$ to the terminal category $1$, they get a magnitude category $RSet_g$ of positive real sets. These are like sets, but their cardinality can be anything in $[0,\infty]$!

For example, Janelidze and Street construct a positive real set of cardinality $\pi$. Unfortunately they do it starting from the binary expansion of $\pi$, so it doesn’t connect in a very interesting way with anything I know about the number $\pi$.

What’s that little subscript $g$? Well, unfortunately $RSet_g$ is a groupoid: the only morphisms between positive real sets we get from this construction are the isomorphisms.

So, there’s a lot of great stuff here, but apparently a lot left to do.

Digressive Postlude

There is more to say, but I need to get going — I have to walk 45 minutes to Paris 7 to talk to Mathieu Anel about symplectic geometry, and then have lunch with him and Paul-André Melliès. Paul-André kindly invited me to participate in his habilitation defense on Monday, along with Gordon Plotkin, André Joyal, Jean-Yves Girard, Thierry Coquand, Pierre-Louis Curien, George Gonthier, and my friend Karine Chemla (an expert on the history of Chinese mathematics). Paul-André has some wonderful ideas on linear logic, Frobenius pseudomonads, game semantics and the like, and we want to figure out more precisely how all this stuff is connected to topological quantum field theory. I think nobody has gotten to the bottom of this! So, I hope to spend more time here, figuring it out with Paul-André.

Posted at November 23, 2017 6:52 AM UTC

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Re: Real Sets

Do you know of a naturally-occurring groupoid whose cardinality is $\pi$? (Or better, $2\pi$?)

Posted by: Tom Leinster on November 24, 2017 3:04 AM | Permalink | Reply to this

Re: Real Sets

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

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

Posted by: John Baez on November 24, 2017 5:35 PM | Permalink | Reply to this

Re: Real Sets

$\pi$ 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.

Posted by: Mark Meckes on November 25, 2017 4:52 PM | Permalink | Reply to this

Re: Real Sets

Mark wrote:

$e$, on the other hand, clearly deserves to be the cardinality of something natural.

As of course it is (for those not already in the know).

$2\pi$ really deserves to be the measure of the unit circle, and insofar as measures are a generalization of cardinalities, maybe this is all we’ll get. It would be nice if we could just stare at a Lie group and read off its total measure, but apart from semisimple ones I don’t know how to do this. (The semisimple ones have a nondegenerate Killing form on their Lie algebra, which equips the group with a measure. For the circle the Killing form is degenerate.)

Posted by: John Baez on November 26, 2017 1:32 AM | Permalink | Reply to this

Re: Real Sets

The Greek letter $\tau$ is sometimes used as a shorthand for $2\pi$.

Re: Real Sets

The decategorification functor $FinSet_g \to \mathbb{N}$ is a bijection on connected components, i.e. up to non-canonical isomorphism there is exactly one finite set of each cardinality. Is the same true for $RSet_g \to [0,\infty]$? If so, then $RSet_g$ could equivalently be defined by specifying the “symmetric group on $x$ letters” for all $x\in [0,\infty]$.

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? It would be disappointing if the notion of “set with real-number cardinality” were not a faithful generalization of the well-known notion of “set with natural number cardinality”.

Posted by: Mike Shulman on November 24, 2017 11:03 AM | Permalink | Reply to this

Re: Real Sets

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.

Posted by: John Baez on November 24, 2017 6:13 PM | Permalink | Reply to this

Re: Real Sets

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?

Posted by: Mike Shulman on November 24, 2017 10:09 PM | Permalink | Reply to this

Re: Real Sets

Whoops! You’re right. I think it should be pretty easy to see, straight from the definition of $RSet_g$, that these extra automorphisms exist. For example,

$1 = h1 + h1$

where the right is short for a countably infinite sum with all but the first two terms zero, and there’s a morphism that switches the two copies of $h1$ (see equation (5.4) in the paper), and I think this morphism is not the identity (since $RSet_g$ is the free magnitude category on the terminal category).

You may not like this, but it’s part of a consistent philosophy about what real sets should be like. When you can chop something in half, you can switch the two parts.

Posted by: John Baez on November 25, 2017 12:11 PM | Permalink | Reply to this

Re: Real Sets

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.

Posted by: Mike Shulman on November 25, 2017 9:01 AM | Permalink | Reply to this

Re: Real Sets

Mike wrote:

Your description also seems to imply that the automorphism groups of every object of $RSet_g$ except $0$ and $\infty$ are isomorphic?

That sounds right. Maybe I’ll ask Street to confirm this and also say exactly what those groups are.

Posted by: John Baez on November 25, 2017 12:15 PM | Permalink | Reply to this

Re: Real Sets

Street said he does not know this characterization of $RSet_g$, but he seemed interested in working it out, since he’s giving a talk on this subject on December 12th.

Posted by: John Baez on November 26, 2017 1:36 AM | Permalink | Reply to this

Re: Real Sets

I think that people are too cavalier with the word “Set” in these contexts. If sets come to model the idea of a collection of objects, these things are too far gone from being called “sets”. Sure, it’s interesting, and it might be useful from a combinatorial point of view. But it seems to me that the only reason that these objects are called sets is to “have a set with half an element”. Which is not a good enough reason to name something “set”.

Posted by: Asaf Karagila on November 27, 2017 5:08 PM | Permalink | Reply to this

Re: Real Sets

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 ${\sqrt {-a}}$ 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 ${\sqrt {-a}}$.

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.

Posted by: John Baez on November 28, 2017 12:09 AM | Permalink | Reply to this

Re: Real Sets

I have to walk 45 minutes to Paris 7 to talk to Mathieu Anel about symplectic geometry

This is probably off topic, but could you possibly tell us what you and Anel are talking about?

Posted by: Eugene on November 29, 2017 4:56 PM | Permalink | Reply to this

Re: Real Sets

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:

and the review paper:

Posted by: John Baez on November 30, 2017 8:38 PM | Permalink | Reply to this

Re: Real Sets

David Li-Bland gave an interesting talk about derived Langrangian correspondences back in 2014. The video is here.

I am interested in derived symplectic geometry in the $C^\infty$ setting. As far as I know this hasn’t really been developed. Maybe we should talk…

Posted by: Eugene on November 30, 2017 10:06 PM | Permalink | Reply to this

Re: Real Sets

I’m not so big on this derived stuff: I’d like to learn about it, but someone told me you shouldn’t drink and derive, so…

Posted by: John Baez on November 30, 2017 10:21 PM | Permalink | Reply to this

Re: Real Sets

This is off topic but could you have n-categories where n is not an integer or infinity?

Posted by: Jeffery Winkler on November 29, 2017 11:18 PM | Permalink | Reply to this

Re: Real Sets

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

Posted by: John Baez on November 30, 2017 8:27 PM | Permalink | Reply to this

Re: Real Sets

If

a spectrum should correspond to a ‘$\mathbb{Z}$-groupoid’ (Categorification)

and genuine $G$-spectra are consider to be RO(G)-graded, then perhaps we have a $RO(G)$-groupoid.

Or perhaps following Peter May here a different grading is occurring there

Logically, equivariant cohomology theories really should be graded on $Pic(Ho G \mathcal{S})$,

but then it seems one should consider the cohomology theories rather than spectra as so graded.

Posted by: David Corfield on December 1, 2017 8:07 AM | Permalink | Reply to this

Re: Real Sets

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!

Posted by: Richard Williamson on December 1, 2017 10:43 AM | Permalink | Reply to this

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