nLab relatively k-compact morphism in an (infinity,1)-category

Definition

Definition

For $\kappa$ some cardinal, say a morphism $f : x \to y$ in $C$ is relatively $\kappa$-compact if for all (∞,1)-pullbacks along $h : y' \to y$ to $\kappa$-compact objects, $y'$, the pulled back object $h^* x'$ is itself a $\kappa$-compact object.

Characterization of $(\infty,1)$-toposes

Theorem

A presentable (∞,1)-category $C$ is an (∞,1)-topos precisely if

1. it has universal colimits;

2. for sufficiently large regular cardinals $\kappa$, $C$ has a classifying object for relatively $\kappa$-compact morphisms.

Lemma

In ∞Grpd the relatively $\kappa$-compact morphisms, $X \to Y$, def. are precisely those all whose homotopy fibers

$$X_{y} := X \times_{Y} \{y\}$$

over all objects $y \in Y$ are $\kappa$-small infinity-groupoids.

Proof

We may write $Y$ as an (∞,1)-colimit over itself (see there)

$$Y \simeq {\lim_{\to}}_{y \in Y} \{y\}$$

and then use the fact that ∞Grpd – being an (∞,1)-topos – has universal colimits, to obtain the (∞,1)-pullback diagram

$$\array{ {\lim_{\to}}_{y \in Y} X_y &\stackrel{\simeq}{\to} & X \\ \downarrow && \downarrow \\ {\lim_{\to}}_{y \in Y} \{y\} &\stackrel{\simeq}{\to}& Y }$$

exhibiting $X$ as an $(\infty,1)$-colimit of $\kappa$-small objects over $Y$. By stability of $\kappa$-compact objects under $\kappa$-small colimits (see here) it follows that $X$ is $\kappa$-compact if $Y$ is.

This is due to Charles Rezk. The statement appears as HTT, theorem 6.1.6.8.