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[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.

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    $\begingroup$ I don't have a counterexample off the top of my head, but I do not think ∞-topoi of homotopy dimension $\le n$ are hypercomplete in general (I'm thinking of something along the lines of the étale topos of $\mathrm{Spec}\mathbb{Z}$ should provide a counterexample). $\endgroup$
    – Denis Nardin
    Commented Mar 20, 2018 at 21:09

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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.

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  • $\begingroup$ For the last argument, one could also use Example 7.2.1.4 which says that $\mathcal{S}_{/S^1}$ is of homotopy dimension $\leq 0$ if and only if $S^1$ is a retract of a CW complex of dim. $\leq 0$ in the homotopy category of spaces (which is not the case). $\endgroup$
    – Alexander Körschgen
    Commented Apr 16, 2018 at 8:52
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As a counterexample to the statement at the end of the question, the $\infty$-topos of parameterized spectra is even of homotopy dimension $\le -1$, but not hypercomplete--as Mathieu Anel pointed out to me when I asked him a similar question.

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