Special Numbers in Category Theory

September 17, 2020

Posted by John Baez

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There are a few theorems in abstract category theory in which specific numbers play an important role. For example:

Theorem. Let $\mathsf{S}$ be the free symmetric monoidal category on an object $x$ . Regard $\mathsf{S}$ as a mere category. Then there exists an equivalence $F \colon \mathsf{S} \to \mathsf{S}$ such that:

$F$ is not naturally isomorphic to the identity,
$F$ acts as the identity on all objects,
$F$ acts as the identity on all endomorphisms $f \colon x^{\otimes n} \to x^{\otimes n}$ except when $n = 6$ .

This theorem would become false if we replaced $6$ by any other number.

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The proof is lurking here. The point is that $\mathsf{S}$ is the groupoid of finite sets and bijections, so $hom(x^{\otimes n} , x^{\otimes n})$ is the symmetric group $S_n$ — and of all the symmetric groups, only $S_6$ has an outer automorphism.

If we replaced the free symmetric monoidal category on one object by some higher-dimensional analogues we could create theorems with all sorts of crazy numbers showing up, like 24 or 240, since we could get homotopy groups of spheres.

Still, it’s a surprise when a theorem with purely category-theoretic assumptions has a specific number other than 0, 1, or 2 in its conclusion. We were talking about these on Category Theory Community Server. Here’s one pointed out by Peter Arndt:

Theorem. The only category for which the Yoneda embedding is the rightmost of a string of 5 adjoints is the category $\mathsf{Set}$ .

The proof is here:

Robert Rosebrugh and R. J. Wood, An adjoint characterization of the category of sets, Proc. Amer. Math. Soc. 122 (1994), 409–413.

Here’s another one that Arndt pointed out:

Theorem. There are just 3 possible lengths of maximal chains of adjoint functors between compactly generated tensor-triangulated categories: 3, 5 and $\infty$ .

The proof is here:

Paul Balmer, Ivo Dell’Ambrogio and Beren Sander, Grothendieck-Neeman duality and the Wirthmüller isomorphism, Compositio Mathematica 152 (2016), 1740–1776.

Reid Barton pointed out another:

Theorem. There are just 9 model category structures on $Set$ .

This was mentioned without proof by Tom Goodwillie on MathOverflow and explained here:

Omar Antolín Camarena, The nine model category structures on the category of sets.

Do you know other nice theorems like this: hypotheses that sound like ‘general abstract nonsense’, with a surprising conclusion that involves a specific natural number other than 0, 1, and 2?

Posted at September 17, 2020 2:07 AM UTC

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Re: Special Numbers in Category Theory

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Still, it’s a surprise when a theorem with purely category-theoretic assumptions has a specific number other than 0, 1, or 2 in its conclusion.

Just barely limping across the finish line, the best I could come up with was 3. And although I have two examples where a 3 appears, I’m not sure either of them quite matches your demands:

3 is the smallest value of $n$ for which the theory of $n$ -categories is nontrivial, in the sense that not every weak $n$ -category is equivalent to a strict $n$ -category.
Let $A$ be a full subcategory of $FinSet$ containing at least one set with at least 3 elements. Then the codensity monad of the inclusion $A \hookrightarrow Set$ is the ultrafilter monad.

(This has the following really arresting corollary, due to Lawvere and got by taking $A$ to be the full subcategory on a 3-element set. For a set $X$ , the maps $3^X \to 3$ that are equivariant with respect to the obvious actions of the 27-element monoid $End(3)$ correspond one-to-one with the ultrafilters on $X$ .)

The theorem could be massaged a bit to look more abstract. For instance, instead of saying FinSet, I could have said the free category with finite coproducts on a single object, and instead of saying Set, I could have said the free category with small coproducts on a single object.

And if you insist on having the 3 appear as part of the conclusion rather than the hypotheses, it can be done. The point is that the original theorem becomes false if 3 is replaced by any smaller number. So:

Theorem Let $X$ be a finite set of smallest cardinality such that the codensity monad of the inclusion functor from the full subcategory on $X$ into $Set$ is isomorphic to the codensity monad of $FinSet \hookrightarrow Set$ . Then $X \cong 3$ .

Posted by: Tom Leinster on September 17, 2020 10:37 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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For instance, instead of saying FinSet, I could have said the free category with finite coproducts on a single object, and instead of saying Set, I could have said the free category with small coproducts on a single object.

And would the free category with all coproducts on a single object be a category of classes?

Posted by: Madeleine Birchfield on September 17, 2020 10:49 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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I suppose? One needs an external measure of size, of course. Or a fixed metatheory that can discuss proper classes. It’s actually quite tricky, since to discuss arbitrary coproducts of classes, one needs to specify what an arbitrary family of classes is, indexed by a class, and such a thing can turn out to be… a map of classes with codomain the indexing class. Because classes can’t be elements of a class! So in the same sense that Set has all coproducts, without an ambient strong metatheory (like ZFC), the statement is slightly vacuous (what is an arbitrary small coproduct in a model of ETCS?).

Otherwise one might take some universes, and then reason that way. I’ve not thought about this option, but it seems both a) easier and b) cheating!

Posted by: David Roberts on September 18, 2020 6:25 AM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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Sometimes cheating is justified.

Posted by: John Baez on September 20, 2020 4:37 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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If one is to hide interesting non-category-theoretic mathematical facts about specific numbers in category-theoretic facts about specific numbers, then surely there must be a way to hide Conway and Doyle’s result on divisibility by 3. Or is the 3 there not special after all?

Posted by: L Spice on September 25, 2020 9:07 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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Section 11 of the Conway-Doyle paper remarks that one can adapt the proof for $3$ to any finite $n$ .

Posted by: Todd Trimble on September 26, 2020 3:47 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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There probably should be a way to hide the monster group and monstrous moonshine in category theory.

Posted by: Madeleine Birchfield on September 26, 2020 6:21 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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This is belated but I just realized that, because the intersection of the empty set with itself is empty, the disjoint union of the empty set with itself is also empty; this is probably old news to many. I read that a group is by definition a nonempty set but apparently the empty set is a fine monoid.

Re: Special Numbers in Category Theory

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Both groups and monoids have identity elements. A semigroup doesn’t have to have any elements.

Posted by: Todd Trimble on October 12, 2020 1:17 AM | Permalink | Reply to this

Ivo

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I was pleasantly surprised this grey Monday morning to see our result mentioned in one of my favorite blogs :-)

I know it’s a bit late, but let me contribute another fascinating example:

Theorem There are precisely 7 closed symmetric monoidal structures on the category of $\mathbb{F}_2[\mathbb{Z}_2]$ -modules (but only 3 different possible underlying tensor functors).

If that’s not clear, the above ring is the group algebra of the group with two elements with coefficients in the field with two elements. Note that if instead we take coefficients in any field with characteristic different from two, suddenly there is a proper class of such structures on the module category. For all this and more:

Mark Hovey, Additive closed symmetric monoidal structures on R-modules, Journal of Pure and Applied Algebra 215 (2011) 789–805.

Posted by: Ivo Dell'Ambrogio on October 12, 2020 12:25 PM | Permalink | Reply to this

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Wow, that’s a great result—a worthy addition to our list here! It’s never too late for such a nice result.

Posted by: John Baez on October 12, 2020 5:49 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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John, thanks.

A couple of thoughts:

You make a comment about homotopy groups of spheres being realizable in this framework. Can you outline the construction here?
Come to think of it, can any finite abelian group be realized as a homotopy group of a sphere?

Posted by: Mike on November 26, 2020 6:49 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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1) It’s a long story, one of the most exciting things I’ve ever learned, but I’ll just do an example. The 2-sphere “is” the free $\infty$ -groupoid $X$ on an object $x$ with an invertible 2-morphism

$f: 1_x \Rightarrow 1_x$

So if you’re interested in $\pi_3(S^2)$ you can take this $\infty$ -groupoid, decategorify it down to a 3-groupoid, and look at the group of invertible 3-morphisms

$g: 1_{1_x} \Rrightarrow 1_{1_x}$

in this 3-groupoid. It turns out this group is isomorphic to $\mathbb{Z}$ , so

$\pi_3(S^2) \cong \mathbb{Z}$

And we could replace the numbers 2 and 3 here by any other natural numbers, making the obvious adjustments, and get $\pi_k(S^n)$ .

This stuff is sketched out in more detail in Higher-dimensional algebra and topological quantum field theory, leading up to the end of section 7.

2) Fun question! I bet that $\mathbb{Z}_5$ is not the homotopy group of a sphere, and I’ll pay someone ten bucks if they prove otherwise. But $\mathbb{Z}_{15}$ is a homotopy group of a sphere in several ways:

$\pi_{10}(S^2) \cong \mathbb{Z}_{15}$ $\pi_{10}(S^3) \cong \mathbb{Z}_{15}$ $\pi_{11}(S^4) \cong \mathbb{Z}_{15}$

So, if you take the free $\infty$ -groupoid on a 4-loop and decategorify it down to an 11-groupoid, the group of 11-loops is $\mathbb{Z}_{15}$ .

Posted by: John Baez on November 26, 2020 7:51 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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I’m glad I said $\mathbb{Z}_5$ . I thought maybe $\mathbb{Z}_4$ wasn’t a homotopy group of spheres, but it turns out $\pi_{n+60}(S^n) = \mathbb{Z}_4$ for all sufficiently large $n$ . So you may have to go pretty far out to settle these questions! And I don’t know yet how to prove a finite abelian group is not a homotopy group of spheres.

Posted by: John Baez on November 27, 2020 1:21 AM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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What’s behind your hunch that $\mathbb{Z}_5$ is not a homotopy group of a sphere?

Posted by: Tom Leinster on December 2, 2020 1:16 AM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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I looked at the biggest table of homotopy groups of spheres that I could find, and noticed that $\mathbb{Z}_p$ never shows up when $p$ is an odd prime greater than 3. And I only see $\mathbb{Z}_3$ appearing in a few ways:

$\pi_9(S^2) \cong \pi_9(S^3) \cong \mathbb{Z}_3$

$\pi_{n+13}(S^n) \cong \mathbb{Z}_3 {\; if \; } n \ge 13+2$

$\pi_{n+29}(S^n) \cong \mathbb{Z}_3 {\; if \; } n \ge 29+2$

Apart from these and three instances of $\mathbb{Z}_{15}$ , which I listed above, these are the only homotopy groups of spheres of odd order that I see, apart from the trivial group!

Of course this is not much evidence, given that there are infinitely many natural numbers. But I’ll conjecture that $\mathbb{Z}_p$ never shows up as a homotopy group of sphere when $p$ is a prime greater than 3.

People do know some general results about homotopy groups localized at primes, but apparently nothing powerful enough to settle this conjecture… except by disproving it with a counterexample.

Posted by: John Baez on December 2, 2020 2:24 AM | Permalink | Reply to this

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September 17, 2020