I'm trying to work through what the $(-1)$-truncated morphisms are in $\def\Catinf{\mathcal{C}\!at_\infty} \Catinf$.

BLUF:The correct characterization is that $F : C \to D$ is a (-1)-truncated map of $\infty$-categories iff, on hom spaces, $C(X,Y) \to D(FX, FY)$ is a (-1)-truncated map of spaces whose essential image contains the equivalences.

I've seen it stated, e.g. at nLab, that these should be precisely the full-and-faithful functors.

However, by 5.5.6.15 of Higher Topos Theory, a functor $F : C \to D$ is $(-1)$-truncated iff the diagonal $\Delta : C \to C \times_D C$ is an equivalence (i.e. $(-2)$-truncated).

Consider the model given by simplicially enriched categories, and the special case that $C$ and $D$ are the ordinary categories ${\bf 1} + {\bf 1}$ and ${\bf 2}$ respectively. That is, $C$ is the discrete category with two elements, and $D$ adjoins a single morphism between them.

Since all of the hom-spaces are either empty or the point, these are fibrant objects. Furthermore, $C \to D$ is a fibration on hom-sets, and has the equivalence lifting property. Thus, $F$ is a fibration of the model structure.

Thus, the ordinary pullback computes the homotopy pullback, and it's easy to see that $C \to C \times_D C$ is, in fact, an isomorphism of simpicially enriched categories.

But $C \to D$ is very much not a full functor.

Instead, if I've worked through the details correctly, a functor $F : C \to D$ being $(-1)$-truncated is equivalent to the weaker condition

- $C(X,Y) \to D(FX, FY)$ is a $(-1)$-truncated map of spaces
- If $FX \simeq FY$, then $X \simeq Y$

This includes full-and-faithful functors, but it also includes more general examples.

So I have conflicting information. Is nLab in error? Have I made an error? Have I made some other serious misunderstanding?