Do homotopy categories of finitely (co)complete quasicategories determine categorical equivalences?

Question

Let $F : C \to D$ be an exact functor between (co)fibration categories such that $Ho(F) : Ho(C) \to Ho(D)$ is an equivalence of homotopy categories. Cisinski proved that in this case $F$ is an equivalence.

It was shown by Szumiło that (co)fibration categories and finitely (co)complete quasicategories are equivalent (as fibration categories). So it seems that analogous theorem might be true for quasicategories.

Let $F : C \to D$ be an exact functor (that is, $F$ preserves finite (co)limits) between finitely (co)complete quasicategories such that $Ho(F) : Ho(C) \to Ho(D)$ is an equivalence of homotopy categories. Does this imply that $F$ is a categorical equivalence of quasicategories?

Denis-Charles Cisinski, MR 2746284 Invariance de la $K$-théorie par équivalences dérivées, J. K-Theory 6 (2010), no. 3, 505--546.

Karol Szumiło, Two Models for the Homotopy Theory of Cocomplete Homotopy Theories.

Answer 1

Yes, this follows from the proof of my result that you quoted, essentially because in my argument weak equivalences of cofibration categories are defined as exact functors inducing equivalences on homotopy categories while equivalences of quasicategories are standard categorical equivalences (and these homotopy theories turn out to be equivalent).

More precisely, let $\mathsf{CofCat}$ and $\mathsf{QCat}_\omega$ be categories of cofibration categories and of finitely cocomplete quasicategories. I construct functors $N_f \colon \mathsf{CofCat} \to \mathsf{QCat}_\omega$ and $\mathrm{Dg} \colon \mathsf{QCat}_\omega \to \mathsf{CofCat}$ that are inverse to each other (in a rather weak but sufficient sense). The proof of Lemma 4.10 in my thesis shows that there is a natural weak equivalence between $\mathrm{Ho}N_f\mathcal{C}$ and $\mathrm{Ho}\mathcal{C}$ for any cofibration category $\mathcal{C}$. Combined with Proposition 4.7 this implies that there is also an equivalence between $\mathrm{Ho Dg}\mathcal{C}$ and $\mathrm{Ho}\mathcal{C}$ for any quasicategory $\mathcal{C}$.

Now, let $\mathcal{C} \to \mathcal{D}$ in $\mathsf{QCat}_\omega$ be a functor that induces an equivalence on homotopy categories. It follows that $\mathrm{Dg}\mathcal{C} \to \mathrm{Dg}\mathcal{D}$ is a weak equivalence of cofibration categories and thus $N_f\mathrm{Dg}\mathcal{C} \to N_f\mathrm{Dg}\mathcal{D}$ is a categorical equivalence of quasicategories (since $N_f$ is an exact functor, Theorem 3.3). This last functor is equivalent to $\mathcal{C} \to \mathcal{D}$ in $\mathsf{QCat}_\omega$ (by Proposition 4.7 again) which is therefore also a categorical equivalence.