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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?

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    $\begingroup$ $F$ is $(-1)$-truncated iff (i) each $C(X,Y)\to D(FX,FY)$ is a $(-1)$-truncated map of spaces which (ii) has all isos $FX\xrightarrow{\sim} FY$ in its effective image. $\endgroup$
    – Charles Rezk
    Commented Nov 10, 2019 at 17:37
  • $\begingroup$ Somewhere along the way, I accidentally replaced (-1)-truncated map of hom-spaces with a bastardization of the characterization of (-1)-truncated spaces. I've corrected my post to undo that error. $\endgroup$
    – Questioner
    Commented Nov 10, 2019 at 18:14
  • $\begingroup$ Monomorphisms in $Cat_\infty$ are characterized this way in Secion 5.1 of Ayala-Francis-Rozenblyum. I'm not sure if that's the earliest reference. $\endgroup$
    – Tim Campion
    Commented Nov 10, 2019 at 19:14
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    $\begingroup$ The characterization I gave is basically obvious if you take complete Segal spaces as your model for $Cat_\infty$, btw. $\endgroup$
    – Charles Rezk
    Commented Nov 10, 2019 at 19:27

1 Answer 1

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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.

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  • $\begingroup$ Thanks; it's good to know I'm not completely wrong here. Also, I have (-1)-truncated map in my work, but somewhere along the line mixed that up with the characterization of (-1)-truncated homotopy types being either empty or (-2)-truncated. Thank you for correcting that too before I did anything depending on that mistake! $\endgroup$
    – Questioner
    Commented Nov 10, 2019 at 17:55
  • $\begingroup$ @Questioner Oh, also the second part in your condition is too weak. As noted by Charles Rezk, you should also require that the image of $X \simeq Y$ is the original $FX \simeq FY$. $\endgroup$
    – Valery Isaev
    Commented Nov 10, 2019 at 18:36
  • $\begingroup$ I guess I did do something depending on that mistake! :( $\endgroup$
    – Questioner
    Commented Nov 10, 2019 at 18:53

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