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

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Relatively compact morphisms

Relatively compact morphisms

Definition

Definition

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

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

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, X→YX \to Y, def. are precisely those all whose homotopy fibers

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

over all objects y∈Yy \in Y are κ\kappa-small infinity-groupoids.

Proof

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

Y≃lim → y∈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

lim → y∈YX y →≃ X ↓ ↓ lim → y∈Y{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.

Last revised on March 2, 2019 at 05:07:08. See the history of this page for a list of all contributions to it.

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