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

- The (possibly or even very probably meaningless) notion of $\alpha$-categories for some real number $\alpha$?
- The same idea but for cardinalities higher than $\aleph_0$?
- 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.

cruciallyuseful for anything. (Yet!) $\endgroup$