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September 17, 2020

Special Numbers in Category Theory

Posted by John Baez

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

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

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

This theorem would become false if we replaced 66 by any other number.

The proof is lurking here. The point is that S\mathsf{S} is the groupoid of finite sets and bijections, so hom(x n,x n)hom(x^{\otimes n} , x^{\otimes n}) is the symmetric group S nS_n — and of all the symmetric groups, only S 6S_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 Set\mathsf{Set}.

The proof is here:

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:

Reid Barton pointed out another:

Theorem. There are just 9 model category structures on SetSet.

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

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

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 nn for which the theory of nn-categories is nontrivial, in the sense that not every weak nn-category is equivalent to a strict nn-category.

  • Let AA be a full subcategory of FinSetFinSet containing at least one set with at least 3 elements. Then the codensity monad of the inclusion ASetA \hookrightarrow Set is the ultrafilter monad.

    (This has the following really arresting corollary, due to Lawvere and got by taking AA to be the full subcategory on a 3-element set. For a set XX, the maps 3 X33^X \to 3 that are equivariant with respect to the obvious actions of the 27-element monoid End(3)End(3) correspond one-to-one with the ultrafilters on XX.)

    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 XX be a finite set of smallest cardinality such that the codensity monad of the inclusion functor from the full subcategory on XX into SetSet is isomorphic to the codensity monad of FinSetSetFinSet \hookrightarrow Set. Then X3X \cong 3.

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

Re: Special Numbers in Category Theory

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

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

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

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

Section 11 of the Conway-Doyle paper remarks that one can adapt the proof for 33 to any finite nn.

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

Re: Special Numbers in Category Theory

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

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.

See also: sound of one hand clapping.

Posted by: jackjohnson on October 11, 2020 6:55 PM | Permalink | Reply to this

Re: Special Numbers in Category Theory

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

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 𝔽 2[ 2]\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:

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

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

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