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Jul 27 at 23:07 comment added Alec Rhea I write this as a comment because I am not an expert in these areas, but: 1. Yes, they are more general in the sense that $(\infty,1)$-categories are those $\omega$-categories where all higher morphisms are invertible. 2. You'd have to be more specific about what you mean by 'finer detail'; anything $(\infty,1)$-categories can do $\omega$-categories can do in principle, but the theory of $(\infty,1)$-categories is much more well developed due to us having a better handle on it. 3. Because they're more complicated. 4. This question is pretty incoherent, apples and oranges.
Jul 27 at 22:17 history asked Pan Mrož CC BY-SA 4.0