[All references are wrt to Lurie's "Higher Topos Theory" in its latest online available version (March 10, 2012)]


**Definition 7.2.1.8**:
An ∞-topos $X$ is locally of homotopy dimension $\leq n$ if there exists a collection $\{U\_\alpha\}$ of objects of $X$ which generate $X$ under colimits, such that each $X\_{/U\_\alpha}$ is of homotopy dimension $\leq n$.


**Definition 7.2.1.1**:
An ∞-topos $X$ has homotopy dimension $\leq n$ if enery $n$-connective object $U \in X$ admits a global section $1\_X \to U$ ($1\_X$ being the final object of $X$).


**Question**: Assume that $X$ is locally of homotopy dimension $\leq n$. Is $X$ of homotopy dimension $\leq n$?


My idea to prove this is using the collection of *all* objects in the first definition. However, I do not get a grasp on any connection between the connectivity of objects of $X$ and the connectivity of objects of some slice ∞-topos $X\_{/U}$.


What I actually want to show is that an ∞-topoi which is of homotopy dimension $\leq n$ for some integer $n$ is hypercomplete. Corollary 7.2.1.12 does this for ∞-topoi which are locally of finite homotopy dimension. Its proof lies on Proposition 7.2.1.10 where I do not understand a crucial point ($\phi$ determines a point of $F(X(0))$), so I am unable to adopt this proposition to ∞-topoi of finite homotopy dimension.



Is an ∞-topos of local homotopy dimension $\leq n$ of homotopy dimension $\leq n$?
Let $\mathcal{X}$ denote the $\infty$-topos $\mathcal{S}\_{/S^1}$, whose objects are spaces $X$ with a map $X \rightarrow S^1$. Then $\mathcal{X}$ is generated under colimits by the object given by the base point inclusion $\ast \rightarrow S^1$, and is therefore locally of homotopy dimension $0$. However, it is not of homotopy dimension $0$, because the map $\ast \rightarrow S^1$ has nonempty homotopy fibers but does not admit a section.


