A topos may be thought of as a generalized topological space. Accordingly, the notions of
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, such that $Sh_{(\infty,1)}(X)$ is hypercomplete, $Sh_{(\infty,1)}(X)$ 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.
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
Under the term locally constant shape the notion appears in section A.1 of
See also
For related references see
Last revised on June 11, 2020 at 06:30:06. See the history of this page for a list of all contributions to it.