Discrete and concrete objects
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.
Let be a ((∞,1)-sheaf-)-topos. It therefore admits a unique geometric morphism ∞Grpd given by global sections. We say that is -connected if is fully faithful.
More generally, we call a geometric morphism between -toposes connected if its inverse image functor is fully faithful.
An -connected -topos has the shape of the point, in the sense of shape of an (∞,1)-topos.
By a basic property of adjoint (∞,1)-functors, being a full and faithful (∞,1)-functor is equivalent to the unit of being an equivalence
By definition of shape of an (∞,1)-topos this means that has the same shape as ∞Grpd, which is to say that it shape is represented, as a functor , 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.
If an -topos is locally ∞-connected (i.e. has a left adjoint ), then is connected if and only if preserves the terminal object.
This is just like the 1-categorical proof. On the one hand, if is ∞-connected, so that is fully faithful, then by properties of adjoint (∞,1)-functors this implies that the counit is an equivalence. But preserves the terminal object, since it is left exact, so .
Conversely, suppose . Then any -groupoid can be written as , the (∞,1)-colimit over itself of the constant diagram at the terminal object (see the details here). Since and are both left adjoints, both preserve colimits, so we have
Therefore, the counit is an equivalence, so is fully faithful, and is ∞-connected.