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 $\mathbf{H}$ be a ((∞,1)-sheaf-)$(\infty,1)$-topos. It therefore admits a unique geometric morphism $(L\Const\dashv\Gamma)\colon \mathbf{H}\xrightarrow{\Gamma}$ ∞Grpd given by global sections. We say that $\mathbf{H}$ is $\infty$-connected if $LConst$ is fully faithful.

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

Properties

Observation

An $\infty$-connected $(\infty,1)$-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, $LConst$ being a full and faithful (∞,1)-functor is equivalent to the unit of $(LConst \dashv \Gamma)$ being an equivalence

$$Id_{\infty Grpd} \stackrel{\simeq}{\to} \Gamma LConst \,.$$

By definition of shape of an (∞,1)-topos this means that $\mathbf{H}$ has the same shape as ∞Grpd, which is to say that it shape is represented, as a functor $\infty Grpd \to \infty Grpd$, 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 $(\infty,1)$-topos $\mathbf{H}$ is locally ∞-connected (i.e. $LConst$ has a left adjoint $\Pi$), then $\mathbf{H}$ 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 $\mathbf{H}$ is ∞-connected, so that $LConst$ is fully faithful, then by properties of adjoint (∞,1)-functors this implies that the counit $\Pi \circ LConst \to \Id$ is an equivalence. But $LConst$ preserves the terminal object, since it is left exact, so $\Pi(*) \simeq \Pi(LConst(*)) \simeq *$.

Conversely, suppose $\Pi(*)\simeq *$. 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 $LConst$ and $\Pi$ are both left adjoints, both preserve colimits, so we have

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

Therefore, the counit $\Pi \circ LConst \to \Id$ is an equivalence, so $LConst$ is fully faithful, and $\mathbf{H}$ 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