# Contents

## Idea

If we think of an (∞,1)-topos as a generalized topological space, then it being ∞-connected is the analogue of a topological space being (weakly) contractible, i.e. weak-homotopy equivalent to a point.

It is an (∞,1)-categorification of the notion of a topos being connected.

## Definition

Let $H$ be a ((∞,1)-sheaf-)$\left(\infty ,1\right)$-topos. It therefore admits a unique geometric morphism $\left(LConst⊣\Gamma \right):H\stackrel{\Gamma }{\to }$ ∞Grpd given by global sections. We say that $H$ is $\infty$-connected if $\mathrm{LConst}$ is fully faithful.

More generally, we call a geometric morphism between $\left(\infty ,1\right)$-toposes connected if its inverse image functor is fully faithful.

## Properties

###### Observation

An $\infty$-connected $\left(\infty ,1\right)$-topos has the shape of the point, in the sense of shape of an (∞,1)-topos.

###### Proof

By a basic property of adjoint (∞,1)-functors, $\mathrm{LConst}$ being a full and faithful (∞,1)-functor is equivalent to the unit of $\left(\mathrm{LConst}⊣\Gamma \right)$ being an equivalence

${\mathrm{Id}}_{\infty \mathrm{Grpd}}\stackrel{\simeq }{\to }\Gamma \mathrm{LConst}\phantom{\rule{thinmathspace}{0ex}}.$Id_{\infty Grpd} \stackrel{\simeq}{\to} \Gamma LConst \,.

By definition of shape of an (∞,1)-topos this means that $H$ has the same shape as ∞Grpd, which is to say that it shape is represented, as a functor $\infty \mathrm{Grpd}\to \infty \mathrm{Grpd}$, by the terminal object $*$. Hence it has the “shape of the point”.

## Locally ∞-connected and ∞-connected

As in the case of connected 1-topoi, we have the following.

###### Proposition

If an $\left(\infty ,1\right)$-topos $H$ is locally ∞-connected (i.e. $\mathrm{LConst}$ has a left adjoint $\Pi$), then $H$ is connected if and only if $\Pi$ preserves the terminal object.

###### Proof

This is just like the 1-categorical proof. On the one hand, if $H$ is ∞-connected, so that $\mathrm{LConst}$ is fully faithful, then by properties of adjoint (∞,1)-functors this implies that the counit $\Pi \circ \mathrm{LConst}\to Id$ is an equivalence. But $\mathrm{LConst}$ preserves the terminal object, since it is left exact, so $\Pi \left(*\right)\simeq \Pi \left(\mathrm{LConst}\left(*\right)\right)\simeq *$.

Conversely, suppose $\Pi \left(*\right)\simeq *$. Then any $\infty$-groupoid $A$ can be written as $A={colim}^{A}*$, the (∞,1)-colimit over $A$ itself of the constant diagram at the terminal object (see the details here). Since $\mathrm{LConst}$ and $\Pi$ are both left adjoints, both preserve colimits, so we have

$\Pi \left(\mathrm{LConst}\left(A\right)\right)\simeq \Pi \left(\mathrm{LConst}\left({colim}^{A}*\right)\right)\simeq {colim}^{A}\Pi \left(\mathrm{LConst}\left(*\right)\right)\simeq {colim}^{A}*\simeq A.$\Pi(LConst(A)) \simeq \Pi(LConst(\colim^A *)) \simeq \colim^A \Pi(LConst(*)) \simeq \colim^A * \simeq A.

Therefore, the counit $\Pi \circ \mathrm{LConst}\to Id$ is an equivalence, so $\mathrm{LConst}$ is fully faithful, and $H$ is ∞-connected.

Revised on April 25, 2013 15:19:36 by Urs Schreiber (82.169.65.155)