nLab
strongly infinity-connected site

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

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 CC 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 CC is in addition an ∞-local site then it is an ∞-cohesive site.

Properties

Proposition

If CC is a strongly \infty-connected site, then the (∞,1)-sheaf (∞,1)-topos Sh (,1)(C)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 Π:Sh (,1)(C)Grpd\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd preserves finite (∞,1)-products.

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

Examples

and

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