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 be the free symmetric monoidal category on an object . Regard as a mere category. Then there exists an equivalence such that:
- is not naturally isomorphic to the identity,
- acts as the identity on all objects,
- acts as the identity on all endomorphisms except when .
This theorem would become false if we replaced by any other number.
The proof is lurking here. The point is that is the groupoid of finite sets and bijections, so is the symmetric group — and of all the symmetric groups, only 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 .
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 .
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 .
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 for which the theory of -categories is nontrivial, in the sense that not every weak -category is equivalent to a strict -category.
Let be a full subcategory of containing at least one set with at least 3 elements. Then the codensity monad of the inclusion is the ultrafilter monad.
(This has the following really arresting corollary, due to Lawvere and got by taking to be the full subcategory on a 3-element set. For a set , the maps that are equivariant with respect to the obvious actions of the 27-element monoid correspond one-to-one with the ultrafilters on .)
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 be a finite set of smallest cardinality such that the codensity monad of the inclusion functor from the full subcategory on into is isomorphic to the codensity monad of . Then .