# Generalizing $n$ for $n$-categories

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.

• There are various reasons to argue that the $n$ in $n$-category is closer to the $n$ in $n$-manifold, and I'm not aware of a really useful generalization of the dimension of a manifold beyond maybe integers. (Hausdorff dimension is I think quite different.) Commented Aug 26, 2022 at 20:02
• @Qiaochu Yuan Well, there's an argument to be made that that continuous geometry did not end up being crucially useful for anything. (Yet!)
– lanf
Commented Aug 26, 2022 at 20:33

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...
• A precise connection between (1) and (2) (possibly mentioned at the discussion you link to) is that abelian group objects in the category of strict $\mathbb{Z}$-categories are equivalent to unbounded chain complexes. Commented Aug 26, 2022 at 21:37
• @Mike : Indeed, nice point ! Maybe even better, the Strict $\mathbb{Z}$-categories whose arrows of all dimension are all strictly invertible should also identifies with unbounded chain complex ( I haven't checked it in details, but that should be true for the same reason as why strict $\infty$-groupoids identifies with crossed complex) ncatlab.org/nlab/show/crossed+complex#from_strict_groupoids Commented Aug 26, 2022 at 23:38