5
$\begingroup$

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?

$\endgroup$
4
  • 1
    $\begingroup$ $F$ 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. $\endgroup$ Nov 10 '19 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
    Nov 10 '19 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
    Nov 10 '19 at 19:14
  • 1
    $\begingroup$ The characterization I gave is basically obvious if you take complete Segal spaces as your model for $Cat_\infty$, btw. $\endgroup$ Nov 10 '19 at 19:27
4
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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
    Nov 10 '19 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 $FX \simeq FY$. $\endgroup$ Nov 10 '19 at 18:36
  • $\begingroup$ I guess I did do something depending on that mistake! :( $\endgroup$
    – Questioner
    Nov 10 '19 at 18:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.