I have two questions:
Is there a way to tell if a functor $F:C \to D$ between two small categories is an equivalence in terms of the map $$N(F):N(C) \to N(D)$$ between simplicial sets? More generally, can we test separately when a functor $F$ is: essentially surjective, full, faithful etc., by an easy to verify property of $N(F)$? I am not interested in any answer involving applying the left-adjoint to $N;$ I am really looking for a description using simplicial sets.
If $\varphi:X \to Y$ are quasi-categories, is there way of saying when $\varphi$ is a weak equivalence in the Joyal model structure (categorical equivalence in the language of Lurie) akin to "full and faithful and essentially surjective?". I am most interested in definitions which are easily checkable, so not definitions like "the induced map on mapping complexes..." since these are hard to compute in practice. The definition I have seen just says that $\varphi$ becomes a weak equivalence of simplicial categories after applying the left-adjoint to the homotopy coherent nerve, which is somehow not a very satisfying definition. Is there a more explicit description, not involving simplicial categories?
I hope this question is not too vague. Any comments to improve the wording etc. are welcome. Thanks!
