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.
A presentable (∞,1)-category $C$ is an (∞,1)-topos precisely if
it has universal colimits;
for sufficiently large regular cardinals $\kappa$, $C$ has a classifying object for relatively $\kappa$-compact morphisms.
In ∞Grpd the relatively $\kappa$-compact morphisms, $X \to Y$, def. are precisely those all whose homotopy fibers
over all objects $y \in Y$ are $\kappa$-small infinity-groupoids.
We may write $Y$ as an (∞,1)-colimit over itself (see there)
and then use the fact that ∞Grpd – being an (∞,1)-topos – has universal colimits, to obtain the (∞,1)-pullback diagram
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.
Last revised on March 2, 2019 at 05:07:08. See the history of this page for a list of all contributions to it.