nLab strongly infinity-connected site

Context

$(\infty,1)$-Topos Theory

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

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

Definition

Definition

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

Remark

If $C$ is in addition an ∞-local site then it is an ∞-cohesive site.

Properties

Proposition

If $C$ is a strongly $\infty$-connected site, then the (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(C)$ over it is a strongly ∞-connected (∞,1)-topos.

Proof

We need to check that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves finite (∞,1)-products.

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

Examples

and

Revised on January 6, 2011 11:22:10 by Urs Schreiber (89.204.137.112)