# Monomorphisms in $\mathcal{C}\!at_\infty$

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?

• $F$ 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. Nov 10 '19 at 17:37
• 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. Nov 10 '19 at 18:14
• 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. Nov 10 '19 at 19:14
• The characterization I gave is basically obvious if you take complete Segal spaces as your model for $Cat_\infty$, btw. Nov 10 '19 at 19:27

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.

• 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! Nov 10 '19 at 17:55
• @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 $FX \simeq FY$. Nov 10 '19 at 18:36
• I guess I did do something depending on that mistake! :( Nov 10 '19 at 18:53