Is there a reason we consider [$\infty$-categories](https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category) to be the $\omega^{th}$ step in the 2-internalization inside **Cat** (or enrichment over **Cat** if you prefer)\* process made invertible above some finite ordinal, and don't continue on to higher steps in the recursion? Is there nothing to be gained, or is the $\omega^{th}$ step already mysterious enough that going further is foolhardy?


For example, it seems (very naively) that something like a $(\omega\_1,\omega)$-category or higher categories defined up to large cardinals that become invertible at smaller large cardinals might be interesting, or in a $\neg CH$ universe we could ask about $(\omega\_1,\mathfrak{c})$-categories and the like. My apologies if this question is trivial, but I couldn't find a discussion/explanation in the literature.




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\*This is an incorrect characterization of how to arrive at a 'fully weak' $\infty$-category (thanks Mike for catching the error), and it appears as though it's an open question wether we can give an algebraic definition of a fully weak $\infty$-category. For details on how to correctly iterate internalization to arrive at a correct definition for weak $n$-categories for all $n$, see [this excellent paper](https://arxiv.org/pdf/1707.01868.pdf) by Simona Paoli.



Higher $\infty$-categories
Using Street's "one type" definition of strict $\infty$-category one can see that the concept of "strict $P$-category" makes sense not justs for any ordinals $P$ but in fact for any posets $P$ (though I expect the definition below is not right when $P$ is not totally ordered, see the remark at the end)


**Definition** : a $P$-category is a set $X$ endowed with:


* (source and target) For each $p \in P$, two functions $\pi\_p^+$ and $\pi\_p^{-}$ from $X$ to $X$.
* (composition) For each $p \in P$ a partially defined composition operation $\#\_p : X \times X \rightarrow X$.


Satisfying the following conditions:


* (globularity relations) For any $q \leqslant p$ one has $ \pi\_p^{\mu} \pi\_q^{\epsilon} = \pi\_q^{\epsilon}$ and if $q>p$ one has $ \pi\_p^{\mu} \pi\_q^{\epsilon} = \pi\_p^{\mu}$.
* (dimension axiom) For any $x \in X$, there exists $p \in P$ such that $\pi^+\_p(x)=x$.
* (domain of composition) $x \#\_p y$ is defined if and only if $\pi^+\_p x= \pi^-\_p y$.
* (boundaries of compositions) $\pi\_p^+(x \#\_p y)=\pi\_p^+ y$ ; $\pi\_p^-(x \#\_p y)=\pi\_p^- x$ and $\pi^{\epsilon}\_q(x \#\_p y) = \pi^{\epsilon}\_q(x) \#\_p \pi^{\epsilon}\_q(y)$ if $q>p$.
* (unit law) For any $x \in X$ and any $n$ one has $x \#\_n \pi\_n^+ x= x = \pi^-\_n x \#\_n x$.
* (associativity of compositions) $(x \#\_p y )\#\_p z = x \#\_p (y \#\_p z)$ when either side is defined.
* (exchange law) for $p< q$, $(w \#\_q x) \#\_p (y \#\_q z) = (w \#\_p y) \#\_q (x \#\_p z)$ when the left hand side is defined.


(where $\epsilon$ and $\mu$ denotes arbitrary signs)


Saying that a cell is ``invertible'' makes sense exactly as in strict $\infty$-category, so you can talk about $(\alpha,\beta)$ categories for any ordinals $\alpha$ and $\beta$.


Now:


* So far we are lacking motivation (interesting examples) to develop such a theory.
* weak version of this notion have not been defined. Moving from strict $\infty$-categories to weak $\infty$-categories is a difficult jump, so there might be a lot of work to came up with such a notion. And as I said we don't really have a motivation to do so.
* Even in the strict case, the "homotopy theory" of such object has not been developed (I'm talking about an analogue of the [Folk model structure](https://ncatlab.org/nlab/show/model+structure+on+strict+omega-categories)). This is probably easier than the point above, but still would need to be done.


As I said we are lacking examples, so it is not clear this notion has any interest at all, but I don't think the notion is trivial in any sense, and actually we do have a few notable examples beyond $\infty$-categories:


For example a $2 \omega$-category with only one cell of dimension $<\omega$ is exactly the same as a strict $\infty$-category with a commutative monoide structure, and I expect that the weak version should be a $E\_{\infty}$-monoidal $\infty$-category in the same way that a weak $\infty$-category
 with only one cell of dimension $<k$ is the same as a $E\_{k}$-monoidal $\infty$-category


As mentioned by Mike above, $\mathbb{Z}$-groupoids are closely connected to spectra. I believe using a similar argument to the Dold-Kan equivalence for strict $\infty$-category one can show that a strict $\mathbb{Z}$-groupoids is the same as an unbounded chain complexes (not quite sure about this... maybe there are problems in $-\infty$ and only a special type of $\mathbb{Z}$-groupoids needs to be considered). If true this is definitely a good indicator that a weak $\mathbb{Z}$-groupoids should be a spectra.


(In fact I remember seeing an arxiv preprint using $\mathbb{Z}$-categories to model spectra or something in this spirit... but I did not really read it and I havn't been able to find it back.)


To me the observation about infinite dimensional sphere is not really relevant here: to me what it says is that spaces don't have information past $\omega$, but this is just the fact that spaces up to homotopy are $\omega$-groupoids and not $\omega^+$-groupoids or anything like this. 


Also if I'm correct the $\omega^+$-category freely generated by an arrows $\theta$ of dimension $\omega$ has for arrows $\theta$ and all the $\pi^{\epsilon}\_n \theta$ for $ n < \omega$ and $\epsilon=+/-$ and that they are all different, so the type of collapse Harry mentioned in his answer do not seem to happen, at least with this definition.


Finally, I believe the definition given here is only right when $P$ is totally ordered, but I suspect that there is a modification of this definition, which is equivalent for totally ordered set, and such that for example a $\mathcal{P}(\{1,\dots,n\})$-category is the same as an $n$-fold category.


