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



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> **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.
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I've seen it stated, e.g. [at nLab](https://ncatlab.org/nlab/show/fully+faithful+%28infinity%2C1%29-functor), 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?



Monomorphisms in $\mathcal{C}\!at_\infty$
The statement at nLab is indeed incorrect, but your condition is also too strong. The first part should be replaced with a weaker condition that the map $C(X,Y) \to D(FX,FY)$ is a $(-1)$-truncated map of spaces.


