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I am reading Riehl's notes "Higher Category from scratch". I have come across the notion of a homotopy 2-category. I have not taken any course on Homotopy Theory before but I know that the homotopy of morphisms in an ordinary category gives rise to 2-cells and we can regard this as an $(\infty,1)$-category.

My question is, when they mention "Each variety of $\infty$-categories will have their own homotopy 2-category", does this mean these $\infty$-category can be thought of as a $2$-catgeory somehow?

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  • $\begingroup$ No, it only means that there is a functor from $\infty$ categories to $2$ categories. This functor loses information, so you can't think of an $\infty$ category as a $2$ category. $\endgroup$
    – Phil Tosteson
    Commented Apr 10 at 14:45
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    $\begingroup$ The "varieties" mentioned here are the $\infty$-cosmoi: well-behaved collections of $\infty$-categories. Intuitively, we have a collection of $\infty$-categories, and for every pair of $\infty$-categories, an $\infty$-category of functors between them. So, these $\infty$-cosmoi should form "$(\infty,2)$-categories." This is analogous to the fact that the category of categories can be thought of as a $2$-category, rather than just an ordinary category. The homotopy $2$-category is a way of analyzing this $(\infty,2)$-categorical situation by means of ordinary $2$-categories. $\endgroup$
    – Stahl
    Commented Apr 10 at 21:01

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