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Let $F:\mathscr{C}\rightarrow \mathscr{D}$ be a 3-functor between 3-categories. Are the following two properties known to be equivalent?

  1. $F$ is a 3-equivalence, meaning that there is a 3-functor $G:\mathscr{D\rightarrow\mathscr{C}}$ and natural equivalences $GF\Rightarrow Id$, $Id\Rightarrow FG$.

  2. $F$ is essentially surjective, and for every object $A,B$ of $\mathscr{C}$, $F$ induces a 2-equivalences of 2-categories $\mathscr{C}(A,B)\rightarrow \mathscr{D}(F(A),F(B))$.

Let me add that 1. might not be exactly the statement one wants. More precisely, maybe we need to specify the inverses of the natural equivalence?

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    $\begingroup$ Regarding your last comment, this might be the difference between an adjoint equivalence, and one where the various data are not chosen to be coherent, or even only postulated to exist. I would start with Nick Gurski's book Coherence in Three-Dimensional Category Theory, or other of his work. The paper Biequivalences in tricategories is not enough, but might give you hints: tac.mta.ca/tac/volumes/26/14/26-14abs.html $\endgroup$
    – David Roberts ♦
    Commented Apr 26, 2020 at 6:25
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    $\begingroup$ That said, I would guess the theorem is true, for the correct definitions of all the things you mention. The trick is more getting the definition completely written down. $\endgroup$
    – David Roberts ♦
    Commented Apr 26, 2020 at 6:26
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    $\begingroup$ There should be such a theorem for $n$-categories for any $n$ (with condition 1 exactly as you stated), in fact even for $(\infty,n)$-categories. There are several equivalent definitions of the latter, and I think this theorem is more or less straightforward using Barwick's complete $n$-fold Segal spaces (see section 14 in arxiv.org/pdf/1112.0040.pdf for the definition). $\endgroup$
    – Marc Hoyois
    Commented Apr 26, 2020 at 8:26
  • $\begingroup$ By "3-category", "3-functor", etc. do you mean weak ones or strict ones? Traditionally in "low-dimensional higher category theory" the words "2-category" and "3-category" mean the strict versions, with "bicategory" and "tricategory" for the weak ones. $\endgroup$
    – Mike Shulman
    Commented May 5, 2020 at 3:26

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