In "Higher Topos Theory", Lurie introduces three different notions of dimension for an $\infty$-topos $\mathcal{X}$, namely:

**Homotopy dimension**(henceforth h.dim.), which is $\leq n$ if $n$-connective objects admit global sections.**Local Homotopy dimension**$\leq n$ if there exist objects $\{ U_\alpha \}$ generating $\mathcal{X}$ under colimits such that $\mathcal{X}_{/U_\alpha}$ is of h.dim. $\leq n$.**Cohomological dimension**(coh.dim.) $\leq n$ if for $k>n$ and any abelian group object $A \in \operatorname{Disc}(\mathcal{X})$, we have $\operatorname{H}^k(\mathcal{X},A) = 0$.

Corollary 7.2.2.30 shows that if $n \geq 2$, and $\mathcal{X}$ is an $\infty$-topos that has finite h.dim. and coh. dim. $\leq n$, then it also has h.dim. $\leq n$. While the converse (h.dim $\leq n$ then also coh.dim. $\leq n$) always holds, the extra requirements are definitely necessary for the given proof; and there is even a counterexample given in 7.2.2.31 for an $\infty$-topos that is of coh.dim. 2, but has infinite h.dim.:

Let $\mathbb{Z}_p$ be the p-adic integers regarded as a profinite group. The example is constructed by forming an ordinary category $\mathcal{C}$ of the finite quotients $\{ \mathbb{Z}_p/{p^n \mathbb{Z}_p}\}_{n \geq 0}$, equipping it with a Grothendieck topology where any nonempty sieve ist covering, and forming the (evidently 1-localic) $\infty$-topos $\mathcal{X}=Shv(N\mathcal{C})$. While I don't completely understand the p-adic methods used in the proof that this is of infinite homotopy dimension, the gist is the following: An $\infty$-connective morphism $\alpha$ in $\mathcal{X}$ ist constructed and it is shown that $\alpha$ can't be an equivalence, so that $\mathcal{X}$ is not hypercomplete and, due to Corollary 7.2.1.12, can therefore not be of **locally** finite homotopy dimension. This is where my issue with the proof lies: Locally finite homotopy dimension does not imply finite homotopy dimension, neither the other way around:

- In Post #80 here, Marc Hoyois gives an example of a cohesive (therefore also finite h.dim.) $\infty$-topos that is not hypercomplete, and can't be locally of finite h.dim because of this. Further, I was told that sheaves over Spectra, e.g. of $\mathbb{Z}$, with the étale topology often also are counterexamples of this direction.
- I unfortunately do not know an example of an $\infty$-topos that is of finite local h.dim. but not of finite h.dim.; I would be happy if anyone could think of one.

It seems to me that this proof in HTT is not complete because of this, and therefore I wanted to ask whether I just didn't properly understand the argument, a part is missing or if the example maybe doesn't even work at all.