# nLab totally infinity-connected site

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

A totally $\infty$-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 $\infty$-connected site if it is also a cofiltered (∞,1)-category.

## Properties

###### Proposition

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

###### Proof

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

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

## Examples

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