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

Definition

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

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. 1 are precisely those all whose homotopy fibers

$$X_y:=X{\times }_Y\{y\}$$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)

$$Y\simeq {\underrightarrow{\lim }}_{y\in Y}\{y\}$$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

$$\begin{array}{ccc}{\underrightarrow{\lim }}_{y\in Y}{X}_y& \stackrel{\simeq }{\to }& X\\ \downarrow & & \downarrow \\ {\underrightarrow{\lim }}_{y\in Y}\{y\}& \stackrel{\simeq }{\to }& Y\end{array}$$\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 $(\mathrm{\infty },1)$-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.