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I've been reading about higher category theory and am curious about the distinctions and relationships between ω-categories and (∞,1)-categories. Specifically, I have the following questions:

  1. Generalization and Complexity:

Are ω-categories more general than (∞,1)-categories due to their ability to express detailed morphism interactions at all levels without requiring invertibility? How does this impact their expressiveness and complexity compared to (∞,1)-categories, which focus on homotopy equivalences?

  1. Finer Detail Capture: Do ω-categories allow for the capture of more fine-grained detail that is lost in the homotopical approach of (∞,1)-categories? If so, in what contexts does this finer detail become significant?

  2. Applications and Understanding: Why are ω-categories considered more challenging and less well-understood than (∞,1)-categories? Are there specific areas of mathematics where the richer structure of ω-categories provides substantial advantages over (∞,1)-categories?

  3. Relationship to Set Theory: How do ω-categories relate to set theory in terms of generality and expressiveness? Do they subsume set theory, or are they specifically tailored for different mathematical structures?

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    $\begingroup$ 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. $\endgroup$
    – Alec Rhea
    Commented Jul 27 at 23:07

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