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Stream: theory: mathematics

Topic: size and dimension

Matteo Capucci (he/him) (Feb 14 2025 at 15:17):

has anyone philosophized on the relation between set-theoretic size and homotopical dimension? specifically, it seems that class-sized objects tend to also have homotopical dimension $> 1$, and so on. the practical consequence is that whenever you find yourself contemplating a large object, it is usually a category/groupoid in some way (i.e. the objects have an informative associated notion of equality/map).
conversely, but perhaps unrelatedly, homotopical dimension is (set-theoretically speaking) about the set-theoretic size of identity sets: has anyone linked the usual hierarchies of cardinals with a more complex, bidimensional, hierarchy of 'cardinals of dimension n' which classify size at the n-th homotopy level? (e.g. $B^n k$ for $n \in \N$ and $k$ finite cardinal would be the 'k-th n-th dimensional' cardinal)

David Corfield (Feb 14 2025 at 15:36):

Are you thinking about [[groupoid cardinality]]?

There was a new paper on that just very recently. See if I can find it.

David Corfield (Feb 14 2025 at 15:39):

It was this article

• Andrés Ortiz-Muñoz, Homotopical Entropy

Matteo Capucci (he/him) (Feb 14 2025 at 16:11):

David Corfield said:

Are you thinking about [[groupoid cardinality]]?

No, here I'm thinking of objective cardinality, as in [[cardinal]]

Matteo Capucci (he/him) (Feb 14 2025 at 16:12):

Though it'd be very cool if 'homotopy cardinals' turn out being the positive rationals/reals just like 'set cardinals' are the positive integers :astonished:

David Corfield (Feb 14 2025 at 16:39):

If cardinals are isomorphism classes of sets, wouldn't the homotopic version just be equivalence classes of $\infty$-groupoids. But these are just homotopy types, put together via their [[Postnikov towers]].

Then there's a map throwing away loads of information from these to the non-negative reals via homotopy cardinality.

E.g., $|core(FinSet)| = e$. I seem to remember us hunting years ago for a good representative for $\pi$. Not sure we found one.

David Corfield (Feb 14 2025 at 16:52):

Of course, one normally limits oneself to finite Postnikov towers, but I recall some rampant speculation with John Baez that there might be a way to control the homotopy cardinality of the 2-sphere to be the same as its Euler characteristic 2, despite its hlevel being infinite.

One thing that tallied is that the 3-sphere shares the same homotopy groups after 2 (visible via the complex Hopf fibration), so we'd expect $0/2 = |S^3|/|S^2|$ to be the same as $1/|\pi_2(S^2)| = 1/\infty = 0$.

Wild days!

Mike Shulman (Feb 14 2025 at 19:27):

This is definitely an interesting direction to think about, although there are pitfalls in taking it too seriously. For instance, in homotopy type theory without higher inductive types, the only way to produce $n$-types for $n>0$ is by using universes. In particular, therefore, one cannot construct any types living in the zeroth universe that are not sets, or any types living in the first universe that are not 1-types, or more generally any types living in the $n^{\rm th}$ universe that are not $n$-types. (One can show this with a semantic argument: there is a model in which the $n^{\rm th}$ universe contains only $n$-types.)

Once there are higher inductive types, then the homotopy level can blow up, and even the zeroth universe contains types that are not an $n$-type for any finite $n$ such as $S^2$. This is perhaps obvious, but less obvious is that even without HITs, not every set (0-type) lives in the zeroth universe. For example, the type of ordinals in the zeroth universe does not itself belong to the zeroth universe, nor is it equivalent to any type in the zeroth universe, by the Burali-Forti-Girard paradox; but it is a 0-type, since ordinals are rigid (have no nonidentity automorphisms).

In particular, although a certain sort of person is tempted to think that maybe one can escape the paradoxes of set theory by observing that "large objects" are groupoids or categories rather than sets, at least no naive approach to such an idea works, becuase you can reconstruct "large sets" from large categories like this.

Evan Washington (Feb 14 2025 at 20:36):

Was this question perhaps inspired by this paper from Steve Awodey (published version here)? Just such a two-dimensional hierarchy is mentioned in the conclusion.

You might also be interested in What is a Higher-Level Set? from Dimitris Tsementzis

Matteo Capucci (he/him) (Feb 18 2025 at 10:08):

Evan Washington said:

Was this question perhaps inspired by this paper from Steve Awodey (published version here)? Just such a two-dimensional hierarchy is mentioned in the conclusion.

Not really, but thanks for the pointer!
Evan Washington said:

You might also be interested in What is a Higher-Level Set? from Dimitris Tsementzis

Likewise.