### 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 $\mathsf{S}$ 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.

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?

## Re: Special Numbers in Category Theory

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

conclusionrather than thehypotheses, it can be done. The point is that the original theorem becomes false if 3 is replaced by any smaller number. So:TheoremLet $X$ 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 $X \cong 3$.