nLab
infinity-connected (infinity,1)-topos

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

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

Properties

Observation

An $\mathrm{\infty }$-connected $(\mathrm{\infty },1)$-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 $(\mathrm{LConst}⊣\Gamma )$ being an equivalence

$${\mathrm{Id}}_{\mathrm{\infty }\mathrm{Grpd}}\stackrel{\simeq }{\to }\Gamma \mathrm{LConst}.$$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 $\mathrm{\infty }\mathrm{Grpd}\to \mathrm{\infty }\mathrm{Grpd}$, 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 $(\mathrm{\infty },1)$-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 $\Pi (*)\simeq \Pi (\mathrm{LConst}(*))\simeq *$.

Conversely, suppose $\Pi (*)\simeq *$. Then any $\mathrm{\infty }$-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

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

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

• connected topos / ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos