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

Definition

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

Theorem

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

  1. it has universal colimits;

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

Lemma

In ∞Grpd the relatively κ\kappa-compact morphisms, XYX \to Y, def. 1 are precisely those all whose homotopy fibers

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

over all objects yYy \in Y are κ\kappa-small infinity-groupoids.

Proof

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

Ylim yY{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

lim yYX y X lim yY{y} Y \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 XX as an (,1)(\infty,1)-colimit of κ\kappa-small objects over YY. By stability of κ\kappa-compact objects under κ\kappa-small colimits (see here) it follows that XX is κ\kappa-compact if YY is.

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

Revised on March 7, 2012 17:30:41 by Stephan Alexander Spahn (79.227.134.205)