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Following "A Short Course on $\infty$-Categories" by Moritz Groth, is there a general guideline for when weak equivalences should depend only on $\pi_0$ information versus all higher homotopical information?

An example: Groth uses the inner horn simplicial set model of an infinity category. Here two vertices in the homotopy category are isomorphic, if and only if, there are edges between the two that act as inverses up to homotopy, inherently a definition involving no higher homotopy.

Contrasting this, we have simplicialy enriched categories forming a model category where we explicitly require that the map on the homotopy category to be essentially surjective in the homotopy category $\textit{and}$ that the map on Hom-sets is a weak equivalence. This specifically invokes "higher homotopical information" rather than just $\pi_0$ information.

Is it correct to think that when "higher homotopical information" is used, somehow we are thinking about the homotopy theory of homotopy theories? (For example, a weak equivalence of spaces uses higher homotopical information, and we can think of this as a map of $\infty$-groupoids). I think this question is essentially equivalent to when homotopy commutativity is enough versus homotopy coherence.

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This gets at some interesting issues, but I think your thoughts may be aimed at a red herring right now. There is, after all, an $\infty$-category whose objects are simplicial categories and whose equivalences are the Dwyer-Kan equivalences.

Perhaps more concretely, a big part of the benefit of model categories is that every weak equivalence is equivalent to a homotopy equivalence, and in particular every weak equivalence of bifibrant objects is a homotopy equivalence. So there is always a "$\pi_0$-based" characterization of weak equivalences in your sense, even though the weak equivalences may not be defined in that way.

What might be a more fruitful direction of thought is: what $\infty$-categorical phenomena can be detected, like isomorphisms, in the homotopy category? For instance, (co)limits rarely can be, through (co)products and terminal (initial) objects are an exception.

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