nLab
totally infinity-connected site

Context

$(∞, 1)$ -Topos Theory

Contents

Idea
Definition
Properties
Examples
Related concepts

Idea

A totally $∞$ -connected site is a site satisfying sufficient conditions to make the (∞,1)-sheaf (∞,1)-topos over it a totally ∞-connected (∞,1)-topos.

Definition

Proposition

Let $C$ be a locally and globally ∞-connected site; we say it is a strongly $∞$ -connected site if it is also a cofiltered (∞,1)-category.

Properties

Proposition

If $C$ is a totally $∞$ -connected site, then the (∞,1)-sheaf (∞,1)-topos ${Sh}_{(∞, 1)} (C)$ over it is a totally ∞-connected (∞,1)-topos.

Proof

We need to check that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor $Π : {Sh}_{(∞, 1)} (C) \to ∞ Grpd$ preserves finite (∞,1)-limits.

By the discussion at ∞-connected site we have that $Π$ is given by the (∞,1)-colimit (∞,1)-functor $\lim_{\to} : Func (C^{op}, ∞ Grpd) \to ∞ Grpd$ . On the opposite and therefore filtered (∞,1)-category $C^{op}$ these preserve finite (∞,1)-limits.

Examples

CartSp, ThCartSp.

Related concepts

and

locally connected site / locally ∞-connected site
- connected site / ∞-connected site
- strongly connected site / strongly ∞-connected site
- totally connected site / totally ∞-connected site
local site, ∞-local site
cohesive site, ∞-cohesive site

Revised on January 6, 2011 01:02:38 by Urs Schreiber (89.204.153.69)