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.

  • 3
    $\begingroup$ 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.) $\endgroup$ Commented Aug 26, 2022 at 20:02
  • $\begingroup$ @Qiaochu Yuan Well, there's an argument to be made that that continuous geometry did not end up being crucially useful for anything. (Yet!) $\endgroup$
    – lanf
    Commented Aug 26, 2022 at 20:33

1 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...

  • 2
    $\begingroup$ 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. $\endgroup$ Commented Aug 26, 2022 at 21:37
  • $\begingroup$ @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 $\endgroup$ Commented Aug 26, 2022 at 23:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.