Is there a reason we consider [$\infty$-categories][1] 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][2] by Simona Paoli.


  [1]: https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category
  [2]: https://arxiv.org/pdf/1707.01868.pdf