A topos may be thought of as a generalized topological space. Accordingly, the notions of
locally simply connected space?
locally $2$-connected space
locally $n$-connected space
have analogs for toposes, (n,1)-toposes and (∞,1)-toposes
locally simply connected (2,1)-topos?
locally $n$-connected $(n+1,1)$-topos
locally $\infty$-connected $(\infty,1)$-topos
The numbering mismatch is traditional from topology; see n-connected space. It reads a bit better if we say locally $n$-simply connected for locally $n$-connected, since locally $1$-(simply) connected is locally simply connected, but being locally $n$-simply connected is still a property of an $(n+1,1)$-topos.
A (∞,1)-sheaf (∞,1)-topos $\mathbf{H}$ is called locally $\infty$-connected if the (essentially unique) global section (∞,1)-geometric morphism
extends to an essential geometric morphism $(\infty,1)$-geometric morphism, i.e. there is a further left adjoint $\Pi$
If in addition $\Pi$ preserves the terminal object we say that $\mathbf{H}$ is an ∞-connected (∞,1)-topos.
If $\Pi$ preserves even all finite (∞,1)-products we say that $\mathbf{H}$ is a strongly ∞-connected (∞,1)-topos.
If $\Pi$ preserves even all finite (∞,1)-limits we say that $\mathbf{H}$ is a totally ∞-connected (∞,1)-topos.
In (Lurie, section A.1) this is called an $(\infty,1)$-topos of locally constant shape.
For $\mathbf{H}$ a locally $\infty$-connected $(\infty,1)$-topos and $X \in \mathbf{H}$ an object, we call $\Pi X \in$ ∞Grpd the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of $X$. The (categorical) homotopy groups of $\Pi(X)$ we call the geometric homotopy groups of $X$
Similarly we have:
For $n \in \mathbb{N}$ an $(n+1,1)$-topos $\mathbf{H}$ is called locally $n$-connected if the (essentially unique) global section geometric morphism is has an extra left adjoint.
For $n = 0$ this reproduces the case of a locally connected topos.
The follow proposition gives a large supply of examples.
Let $C$ be a locally ∞-connected (∞,1)-site/∞-connected (∞,1)-site. Then the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(C)$ is a locally $\infty$-connected $(\infty,1)$-topos.
See locally ∞-connected (∞,1)-site/∞-connected (∞,1)-site for the proof.
In (SimpsonTeleman, prop. 2.18) is stated essentially what the above proposition asserts at the level of homotopy categories: if $C$ has contractible objects, then there exists a left adjoint $Ho(\Pi):Ho(Sh_{(\infty,1)}(C)) \to Ho(\infty Grpd)$.
This includes the following examples.
The sites CartSp${}_{top}$ $CartSp_{smooth}$ $CartSp_{synthdiff}$ are locally $\infty$-connected. The corresponding $(\infty,1)$-toposes are the cohesive (∞,1)-toposes ETop∞Grpd, Smooth∞Grpd and SynthDiff∞Grpd.
For $X$ a locally contractible space, $Sh_{(\infty,1)}(X)$ is a locally $\infty$-connected $(\infty,1)$-topos.
The full subcategory $cOp(X) \hookrightarrow Op(X)$ of the category of open subsets on the contractible subsets is another site of definition for $Sh_{(\infty,1)}(X)$. And it is a locally ∞-connected (∞,1)-site.
By the same kind of argument:
For $n \in \mathbb{N}$ and for $X$ a locally $n$-connected topological space, $Sh_{(n+1,1)}(X)$ is a locally $n$-connected $(n+1)$-topos.
For $X$ a locally contractible topological space we have that the fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos computes the correct homotopy type of $X$:
the image of $X$ as the terminal object in $Sh_{(\inffty,1)}(C)$ under the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor
is equivalent to the ordinary fundamental ∞-groupoid given by the singular simplicial complex
By using the presentations of $Sh_{(\infty,1)}(X)$ by the model structure on simplicial presheaves as discussed at locally ∞-connected (∞,1)-site one finds that this boils down to the old Artin-Mazur theorem. More on this at geometric homotopy groups in an (∞,1)-topos.
For $\mathbf{H}$ a locally $\infty$-connected $(\infty,1)$-topos, also all its objects $X \in \mathbf{H}$ are locally $\infty$-connected, in that their petit over-(∞,1)-toposes $\mathbf{H}/X$ are locally $\infty$-connected.
The two notions of fundamental $\infty$-groupoids of $X$ induced this way do agree, in that there is a natural equivalence
By the general facts recalled at etale geometric morphism we have a composite essential geometric morphism
and $X_!$ is given by sending $(Y \to X) \in \mathbf{H}/X$ to $Y \in \mathbf{H}$.
If in the above $X$ is contractible in that $\Pi X \simeq *$ then $\mathbf{H}/X$ is even an ∞-connected (∞,1)-topos.
By the discussion there we need to check that $\Pi_X$ preserves the terminal object:
Let $\mathcal{X}$ be an $(\infty,1)$-topos and $\{U_i\}_i$ a collection of objects such that
the canonical morphism $\coprod_i U_i \to *$ out of their coproduct to the terminal object is an effective epimorphism;
all the slice-(∞,1)-toposes $\mathcal{X}_{/U_i}$ are locally $\infty$-connected.
Then also $\mathcal{X}$ itself is locally $\infty$-connected.
This appears as (Lurie, corollary A.1.7).
For $(\Pi \dashv \Delta \dashv \Gamma) : \mathbf{H} \to \infty Grpd$ a locally $\infty$-connected $(\infty,1)$-topos, its underlying (1,1)-topos $\tau_{\leq 0} \mathbf{H}$ is a locally connected topos. Moreover, if $\mathbf{H}$ is strongly connected (the extra left adjoint preserves finite products), then so is $\tau_{\leq 0} \mathbf{H}$.
The global sections geometric morphism $\Gamma \simeq \mathbf{H}(*,-)$ is given by homming out of the terminal object and hence preserves 0-truncated objects by definition. Also, by the $(\Pi \dahsv \Delta)$-adjunction so does $\Delta$: for every $S \in Set \simeq \tau_{\leq }\infty Grpd \hookrightarrow \infty Grpd$ and every $X \in \mathbf{H}$ we have
Therefore by essential uniqueness of adjoints the restriction $\Delta|_{\leq 0} \colon Set \hookrightarrow \infty Grpd \stackrel{\Delta}{\to} \mathbf{H}$ has a left adjoint given by
Finally, by the discussion here, $\tau_{\leq 0}$ preserves finite limits. Hence $\Pi_0$ does so if $\Pi$ does.
The fact that the terminal geometric morphism is essential gives rise to various induced structures of interest. For instance it induces a notion of
For a more exhaustive list of extra structures see cohesive (∞,1)-topos.
locally connected topos / locally ∞-connected (∞,1)-topos
and
Some discussion of the homotopy category of locally $\infty$-connected $(\infty,1)$-toposes is around proposition 2.18 of
Undet the term locally constant shape the notion appears in section A.1 of
For related references see