# Link between homotopy equivalence of simplicial sets and categorical equivalences

In Higher Topos Theory, a map $$f: S \rightarrow T$$ of simplicial sets is a categorical equivalence if after applying the functor $$\mathfrak{C}[-]$$ we have an equivalence of simplicial categories.

In the proof of Theorem 2.2.5.1, we see for example that any trivial fibration, in the Kan model structure on simplicial sets, is a categorical equivalence.

My question is the following : under what conditions can we say that a Kan weak equivalence is a categorical equivalence?

I am asking this question because when Lurie finally proves Proposition 1.2.9.3 in section 2.4.5, he claims (I think), at some point that a weak equivalence between two Kan complexes is a categorical equivalence. (This is to use Corollary 2.4.4.4.)

• A map of $\infty$-groupoids (i.e. Kan complexes) is a categorical equivalence if and only if it is a Kan equivalence, and the fibrant replacement map from $\mathcal{C}$ to its 'groupoidifcation' is a Kan equivalence so you learn that a functor is a Kan equivalence if and only if it induces a categorical equivalence after inverting all the arrows. – Dylan Wilson Jan 3 '19 at 17:00

While Dylan's comment is correct, it doesn't really explain how this is shown. It's actually a corollary of the existence of the Joyal model structure, which is proven earlier in the chapter as a corollary of the comparison theorem with simplicial categories.

Since the cofibrations for both the Joyal and Kan-Quillen model structures coincide, and since the fibrant objects for the Kan-Quillen model structure are also fibrant for the Joyal model structure, it follows that the Kan-Quillen model structure is a left-Bousfield localization of the Joyal model structure.

It is a general fact of the theory of (left) Bousfield localization (see e.g. Hirschhorn) that the local equivalences between local objects are exactly the equivalences in the original unlocalized model structure.

I don't see any way to show this directly just from the given description of the weak equivalences without first proving the comparison theorem.

• Well, one can construct the Joyal model structure without proving the comparison theorem. Joyal did this, and so does Cisinski in his new book. But maybe I'm misreading your closing claim. – Kevin Arlin Jan 3 '19 at 20:39
• For example, the claim is relatively straightforward using the definition of categorical equivalence given by Joyal (fully faithful and essentially surjective), or using the version of "weak categorical equivalence". In the former case it's essentially the Whitehead theorem (detecting we's by pi_* isomorphisms) and in the latter it's basically Yoneda. – Dylan Wilson Jan 3 '19 at 21:02
• Yes, both of you guys are right @KevinCarlson , but the definition given in the question of a categorical equivalence was the one I was aiming at. I am well-acquainted with the Joyal and Cisinski approach, but that wasn't the context of the question, for better or for worse. I was trying to be faithful to the flow of the argument in HTT without bringing in other (imo better) sources, since the other definitions are only equivalent by means of the comparison theorem itself. – Harry Gindi Jan 3 '19 at 21:16
• How do we know that every fibrant object in the Kan-Quillen model structure are also fibrant in the Joyal model structure? It is shown in HTT, that every $\infty$-category is fibrant in the Joyal model structure and this would be enough but I have a problem : the proof of this needs Proposition 1.2.9.3 and we would have a circular argument. Am I missing something obvious? – Oscar P. Jan 6 '19 at 10:35
• @OscarP. Do you know exactly where he uses that statement that a categorical equivalence of Kan complexes is a weak homotopy equivalence? I couldn't find it. – Harry Gindi Jan 6 '19 at 17:33