Generalizing $n$ for $n$-categories

Question

I understand that this is a very amateurish question, but perhaps I can be forgiven since higher category theory (or indeed any sort of category theory) is not my field.

I have been thinking about a very superficial similarity between the dimension of a vector space, and the $n$ for $n$-categories. Namely that once it was clear you could generalize this property of an object to any natural number, people very quickly generalized this to $\infty$.

For vector spaces, we have generalizations even beyond this, for example uncountably infinite dimensional spaces, or von Neumann's continuous geometry, which explores an idea vaguely like giving a space a real-valued dimension.

I have two questions.

Can anyone give more examples of this phenomenon, where you have a property of some mathematical object associated with a natural number, and this property is generalized far outside $\mathbb{N}$?

Is there any existing work in higher category theory that explores

1. The (possibly or even very probably meaningless) notion of $\alpha$-categories for some real number $\alpha$?
2. The same idea but for cardinalities higher than $\aleph_0$?
3. The almost certainly meaningless notion of what it would mean to talk about an sequence of categories $C_1, C_2, \dots, C_{n - 1}$, where the $i$ in $C_i$ lives in $\mathbb{Z}_i$? I've given no thought whatsoever to whether this would make sense, but perhaps the $n$-morphisms are somehow also the objects in $C_1$?

Thank you for indulging me.

Answer 1

Here are some things I know of that looks like this:

1. I* have argued here that it is possible to extend the notion of strict $\infty$-category to a notion of $P$-category for any poset $P$. At this point this is more of a curiosity than anything serious: I don't know of any use of this definition, it is not quite clear that this is the correct definition when $P$ is not linearly ordered, and I do not quite know how to make a non-strict version of the notion (I haven't thought about it). But you can definitely take $P = \mathbb{R}$ or $P =\alpha$ for any ordinal $\alpha$ and you would get something (especially since both are linearly ordered). [* : to be clear - I don't think I'm the first person to make this observation - I just haven't seen it written elsewhere.]

2. It is very natural to think about spectra (or rather $\Omega$-spectra) as a kind of weak $\infty$-groupoids with arrow of all dimension $n \in \mathbb{Z}$ instead of $n \in \mathbb{N}$. Indeed, using the representation of spectra as sequences of pointed spaces with equivalence $X_i \simeq \Omega(X_{i+1})$ is saying that the "objects" of the $\infty$-groupoid $X_0$ are themselves arrows $* \to *$ in the $\infty$-groupoids $X_1$, whose object can hence be thought of as "$-1$ arrows" in $X_0$, but objects of $X_1$ themselves identifies with arrows $* \to *$, so objects of $X_2$ can be thought of as $-2$ arrows and so one...

3. Here is an interesting example of such a structure that "appear in nature". It would require too much work to explain in full detail, but maybe I can give a general idea of what it looks like: In Homotopy type theory, every type naturally gets the structure of a weak $\infty$-groupoids (with arrows given by the identity types). But this should also be true "internally" in the type theory (Note: there are big difficulties to make this formal in pure HoTT, but some extension of HoTT can do this for sure), so we get a structure of $\infty$-groupoids with arrows not just indexed by the standard natural numbers, but also by all the term $n$ of type $N$ in the type theory. In the syntactic, or any "standard" model of HoTT, terms of type "N" are just natural numbers ( only conjecturally for the syntactic model maybe ? not sure what is the status of this). But there exists non-standard model of HoTT (They exist by a compactness argument) in which we will get more terms of type $N$ (up to identities of course). The set of all these terms is going to constitute (at least) a non-standard model of Peano arithmetic, so we know that the countable ones have order type $\mathbb{N} + \mathbb{Z} \times \mathbb{Q}$ as every countable non-standard model have this order type. And every type in such a theory will define a kind of $\infty$-groupoids that have arrows of all "dimension" $n \in \mathbb{N} + \mathbb{Z} \times \mathbb{Q}$. A similar story can be told in the theory of $\infty$-topos instead of HoTT, where non-standard elementary $\infty$-toposes have been explicitly constructed here. These have an uncountable number of sections of $N$, so we get a more complicated order type, but still...