higher category theory – Counterexamples concerning $infty$-topoi with infinite homotopy dimension

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 $mathcal{X}=Shv(Nmathcal{C})$. 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 ├ętale 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.