The n-Category Café

Does the zeroth power of a real number count as an example? How about the factorial of 0?

In other words, do mathematical phantoms usually turn out to have at least one valid “solidification” —a consistent definition when seen from a more general point of view, such as “in the limit” or “in agreement with a set of axioms”? That would also allow things like those infinite series that “converge” to values of the Riemann zeta function; we really mean that the definition of that series is taken to be the output of the function.

Hmmm.

Maybe “the sporadic SIC in dimension 24” would fit the bill. In the argot of quantum information theory, a SIC is a set of $d^2$ equiangular lines in $\mathbb{C}^d$. The known examples fall into a mainline family (the “Weyl–Heisenberg SICs”) and a set of sporadic solutions, the SICs in dimensions 2 and 3 as well as one set of those in dimension 8. Among the weird things that happen with the sporadic SICs are unexpected relations to real equiangular lines in nearby dimensions, particularly a link between SICs in $\mathbb{C}^8$ and the maximal set of equiangular lines in $\mathbb{R}^7$. That construction in $\mathbb{R}^7$ is also extractable from the $\mathrm{E}_8$ lattice, just as the maximal set in $\mathbb{R}^{23}$ can be obtained from the Leech lattice. (See arXiv:2008.13288 for details and literature pointers.) So, it would be nice if there were a sporadic SIC up in $\mathbb{C}^{24}$ which, when poked, yielded the maximal set of equiangular lines in $\mathbb{R}^{23}$.

Zhu (2015) proved that there is no doubly transitive SIC in $\mathbb{C}^{24}$. This rules out an exact analogy between the $\mathbb{C}^8 \to \mathbb{R}^7$ case and $\mathbb{C}^{24} \to \mathbb{R}^{23}$, since the sporadic SICs in $\mathbb{C}^8$, the SICs of Hoggar type, are doubly transitive. So, the sporadic SIC in $\mathbb{C}^{24}$ feels more like a “phantom” than merely an unknown. Chatting informally with people who know a lot about this, the sense is that if something unusual happens, it would most naturally happen in dimension 24, but it can’t be the same kind of unusual thing already observed in dimensions 2, 3 and 8.

The symmetry groups of the doubly-transitive SICs relate in a nice way to lattices of integers in $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ (Eisenstein, Hurwitz and Cayley respectively). So, asking for this pattern to continue one step further is a bit like what happens with $\mathrm{G}_3$: There isn’t another real normed division algebra for it to be the automorphism group of.

We have a page for SL(2,O) where Hitchin’s paper is mentioned. Always plenty to add of course.

It’s easy to say which homotopy types contain groups – they are exactly the loop spaces – and harder to say which ones contain manifolds. There’s been a bunch of work on figuring out which contain both; e.g. “A finite loop space not rationally equivalent to a compact Lie group”. Obviously the homotopy type of a Lie group contains both, but it’s interesting to find manifolds that are homotopic to groups but not to Lie groups. IIRC they automatically have trivial tangent bundle, and one can develop a notion of maximal torus for them.

Dyson’s “Missed opportunities” suggests a phantom 26-dim Lie group. Has it been “found” since then?

I did wording that something that exists acts as if it did problematic. Something that doesn’t exist cannot act because it doesn’t exist. So what does the acting? Meaning, of we peel away the “figurative” wording, what could we say that would make better sense?

I was trying to recall a conference I co-organised – Two streams in the philosophy of mathematics. I see I was comparing Alexandre Borovik’s phantoms and Michael Harris’s avatars:

For me two of the most interesting issues to emerge during the conference was Borovik’s ‘phantoms’ and Harris’s ‘avatars’. The first of these may occur when there is a question as to whether a certain entity exists. Even if it does not, it may transpire that some counterpart of this nonexistent entity exists elsewhere. The setting of finite simple groups is a rich environment for this phenomenon.

In the case of avatars, on the other hand, they all exist, but they indicate the existence of a not yet expressible universal object. Grothendieck’s theory of motives is the classic example, and indeed it was here that he coined the term ‘avatar’ to describe an instantiation of a motive in a particular cohomological setting.

Michael’s talk is still available, here.

Another instance, in the penultimate paragraph of this section of his page on a proposal to characterise M-theory, Urs compares construction of the later to the construction of absolute geometry over $\mathbb{F}_1$.

I believe infinitesimals qualify as phantoms, at least in the mainstream approach to calculus. They ‘ought to’ exist, and are even manipulated to informally justify many arguments, but they don’t actually exist since in classical mathematics R is a field implies all infinitesimals are 0. If you work within constructive mathematics though, you can axiomatize fields in such a way that nonzero nilpotent elements cannot be chased away by the axiom of invertibility. Hence there are models of the real numbers with infinitesimals! And in that case, phantoms become very much… real (numbers).

Writing this makes me think: are there any other phantoms which are phantoms of constructive theories (killed by LEM)?

For example, the non-existence of the field with one element also leans on the axiom of invertibility of a field which forces a field to have two elements. Maybe one can define F1 to be the Heyting field where 0 is not apart from 1?

I think you’re onto something with the suggestion that the rigidity of algebra has to do with the plethora of phantoms in that subject. I’ve been trying to coming up with phantoms in analysis and probability, and mostly I’ve come up with things that probably seemed like phantoms at some point in time, but have become perfectly respectable mathematical objects via one kind of completion process or another. Examples would include:

• Irrational numbers (as you mentioned above)
• Distributions, like the Dirac delta function and its derivatives
• Infinitesimals (as Matteo Capucci mentioned above)
• Graph limits

Some other things I’ve thought of are examples of things we would like to exist, but somehow don’t come close enough even to have a phantom existence, like Lebesgue measure on an infinite dimensional Banach space.

One example that might count is infinite dimensional limits of random matrices, which often “exist” as noncommutative random variables in free probability theory. Or maybe those should actually be thought of as avatars rather than phantoms.

I think the uniform probability distribution on the real line is a continuous phantom. It is somewhat resistant to understanding via completion processes. Or maybe someone here knows how to make sense of it.

My favourite phantoms (maybe ex-phantoms now) are non-wellfounded sets. Azcel’s 1988 book brought together the theory of how to get this working, by identifying sets with decorated graphs; Barwise & Moss 1996 give some applications (see here). Conway’s hyperreals are kind of similar.

Recently, some philosophers got interested in impossible worlds: these are metaphysical phantoms. Though the ultimate metaphysical phantom, imo, is the global truth predicate (only “local approximations” exist).

Probably my favorite mathematical phantom is the set of all sets. (Or the (2-)category of all categories, or the type of all types, etc. — I’m talking about size issues, not categorical dimension.)

Here are some more phantoms from the Category Theory Community Server, where my link to this post triggered a lively discussion. Life is getting complicated now, with two active category-oriented forums to attend to.

Nathanael Arkor wrote:

“Small (nonposetal) cocomplete category” has a similar flavour to some of the examples mentioned here (but may actually be reified in a constructive setting). The study of locally presentable categories is in part inspired by this search. Are there any other categorical examples?

Matteo Capucci wrote:

In fact, I’d say pure phantoms do not last long since once you realize you may want them you’re halfway through formalizing them. A different kettle of fish is to formalize them right. E.g., $\mathbb{F}_1$​ has been formalized for at least 25 years but we are still in the pre-paradigmatic phase of the associated Kunnian revolution. Infinitesimals are probably stuck there too.

Amar Hadzhisanovic wrote:

So mathematical phantoms go the other way compared to regular phantoms: they are spirits of the not-born-yet rather than spirits of the dead…

I suppose pure motives are another quintessential mathematical phantom.

Or maybe that should be mixed motives. I don’t really know much about it.

Jules Hedges wrote:

A general class of examples is “things that exist in a conservative extension of your theory”.

For example if you’re working in ZFC then you can pretty much get away with pretending that proper classes exist most of the time, even though plain ZFC doesn’t think they exist.

Morgan Rogers wrote:

Or classifying objects which aren’t representable (things which exist in e.g. a topos associated to your category, but don’t pull back to concrete things in the category).

Peter Arndt wrote:

The machinery of topos theoretic spectra is a midwife helping unborn spirits into existence: E.g. the initial local ring under a given ring in general doesn’t exist … in the category of sets — but it does exist in another topos!

Quantum Field Theory!

I think “space isomorphic with its own function space” provides another nice example of a mathematical phantom, whose non-existence haunted at Dana Scott before he was finally able to prove its existence. He tells this story in a 1976 lecture:

Returning to 1969, what I started to do was to show Strachey that he was all wrong and that he ought to do things in quite another way. He had originally had his attention drawn to the λ-calculus by Roger Penrose and had developed a handy style of using this notation for functional abstraction in explaining programming concepts. It was a formal device, however, and I tried to argue that it had no mathematical basis. I have told this story before, so to make it short, let me only say that in the first place I had actually convinced him by “superior logic” to give up the type-free λ-calculus. But then, as one consequence of my suggestions followed the other, I began to see that computable functions could be defined on a great variety of spaces. The real step was to see that function-spaces were good spaces, and I remember quite clearly that the logician Andrzej Mostowski, who was also visiting Oxford at the time, simply did not believe that the kind of function spaces I defined had a constructive description. But when I saw they actually did, I began to suspect that the possibilities of using function spaces might just be more surprising than we had supposed. Once the doubt about the enforced rigidity of logical types that I had tried to push onto Strachey was there, it was not long before I had found one of the spaces isomorphic with its own function space, which provides a model of the “type-free” λ-calculus. The rest of the story is in the literature.

(Incidentally, the reference to Penrose might be surprising, but I was just listening to a panel discussion about Christopher Strachey where Penrose tells this story of how he managed (after several attempts!) to get Strachey interested in λ-calculus.)

Closely related to infinitesimals suggested by Capucci, my favorite non-algebraic example of a phantom is a nearness relation in topology.

One likes to say that continuous map take nearby values at nearby points. Naively, one could imagine a formalization where a topological space consists of a set S, along with a distinguished binary relation ≈ (nearness), and the continuous maps between topological spaces are just the functions that preserve this relation.

To the regret of Real Analysis 101 students everywhere, this naive formalization does not work: there is no binary relation ≈ on the real numbers such that f: ℝ→ ℝ is continuous precisely if it preserves ≈.

But (as Borovik pointed out) even if a desired entity does not exist, it may transpire that some counterpart of the nonexistent entity exists elsewhere. Classical set theory provides nearness relations for a well-known class, the so-called Alexandroff spaces. For functions between Alexandroff spaces, the specialization preorder ≤ acts as a nearness relation.

Unfortunately, one cannot go further: Top is not equivalent to any full subcategory of the category of (reflexive) graphs, and in fact the category of Alexandroff spaces is maximal with this propery.

However, a bit of non-standard analysis takes us really close: if we replace the notion of binary relation definable in ZFC with binary relation definable in Nelson’s Internal Set Theory, we get nearness relations that work for standard continuous functions between standard topological spaces. I.e. in IST we can uniformly equip each standard topological space S with a relation ≈S such that a standard function f: A→B between two spaces is continuous precisely if x ≈A y implies f(x) ≈B f(y) for all x,y in A.

These new relations are very nice to have: using these, one can write the topological semantics for intuitionistic logic in Kripke style, and present sheaves as pointwise assignments of stalks to points and transition functions to pairs of nearby points (the way Michael Robinson prefers to present sheaves on Alexandroff spaces).

A bit late to the party, but my favorite might be “the space of random sequences” - with, of course, its non-phantasmal counterpart, the locale of random sequences. Alex Simpson’s paper on the topic was a significant influence in getting me interested in locale theory.

The way I have understood ‘the field of order 1’ (to the extent I have understood it, which isn’t great) is in terms of counting formulas.

There isn’t a literal field of order 1, so, in the metric defined on the real numbers, the order $q$ of a finite field has $|q-1| \geq 1$. But that’s not the end of the story. There are other metric completions of $\mathbb{Q}$, one for each prime $p$, which are the $p$-adic completions of $\mathbb{Q}$. The $p$-adic distance $||_{p}$ has a ‘blind spot’, which is that any (positive integral) power of $p$ has distance 1 from 1. But it is possible for powers of other primes to converge to 1 $p$-adically; this can happen in sequences whether or not more than one prime appears this way.

If you fix a single prime $r$ distinct from $p$ when doing that, something nice happens:

Let $o_{p}(r)$ be the order of $r$, considered as an element of $(\mathbb{Z}/(p))^{\times}$. Let $\{ a_n \}$ be a sequence of positive integers. Then the sequence $r^{a_n}$ converges $p$-adically to 1 if and only if $o_{p}(r) \mid a_n$ for all sufficiently large $n$, and the sequence $a_n$ itself converges to 0 $p$-adically. Each $r^{a_n}$ is the order of a characteristic $r$ finite field, and considering such a sequence of characteristic $r$ fields aligns nicely with the description of their algebraic closure as being the direct limit of all finite fields of characteristic $r$.