# ct.category theory – Categories \$ infty \$ higher

Is there a reason we consider $$infty$$-categories of being the $$omega ^ {th}$$ step in internalisation 2 inside Cat (or enrichment on Cat if you prefer) process rendered invertible over a finite ordinal, and do not proceed to the higher stages of recursion? Is there nothing to gain, or is there $$omega ^ {th}$$ already mysterious enough to go further is reckless?

For example, it seems (very naively) that something like a $$( omega_1, omega)$$– a category or higher categories defined up to the great cardinals that become invertible in the greatest cardinals could be interesting, or in a $$neg CH$$ universe we could ask questions about $$( omega_1, mathfrak {c})$$-categories and the like. I apologize if this question is trivial, but I have not found any discussion / explanation in the literature.