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](https://nforum.ncatlab.org/discussion/5347/dcct-to-go-to-the-arxiv/), 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.



Counterexamples concerning $\infty$-topoi with infinite homotopy dimension
To close this thread off, I will try to expand Lurie's helpful comment into an answer:


First of all concerning examples of $\infty$-topoi that are locally, but not globally, of finite homotopical dimension, an easy counterexample is the slice $\infty$-topos $\mathcal{S}\_{/X}$ with $X$ a space that is not a retract of a finite-dimensional CW complex - the fact that this must have infinite homotopy dimension follows directly from HTT 7.2.1.4, which implies that if it were finite, our space $X$ would have to be a retract of a finite-dimensional CW complex.


I also found a different example given around [here](https://chat.stackexchange.com/transcript/message/52490314#52490314), which is very interesting as it shows that even presheaf $\infty$-categories, that are (using that the representable presheaves generate them under colimits) always of local homotopy dimension $\leq 0$, can have infinite homotopy dimension.


Now to the main part of by question, I will try to explain why this works using the definitions I gave above (in particular using only material from HTT); this is a bit more subtle than I thought at first glance, so please correct me if I am talking nonesense here. In his comment, Lurie states a fact that can be interpreted as a kind of *self-similarity* of the $p$-adic numbers: Every finite index subgroup of $\mathbb{Z}\_p$ is itself isomorphic to $\mathbb{Z}\_p$. This allows to show that we can transfer our knowledge that $Shv(N\mathcal{C})$ is of finite homotopy dimension onto the slice categories over representable sheaves - roughly as follows.


Remember that we had chosen the so-called *atomic* Grothendieck topology on $\mathcal{C}$. With a bit of work, we can show that the usual equivalence $\mathcal{P}(N\mathcal{C})\_{/j(C)} \simeq \mathcal{P}(N\mathcal{C}\_{/C})$, for $j$ the Yoneada embedding and $C \in \mathcal{C}$, restricts to $Shv(N\mathcal{C})\_{/j(C)} \simeq Shv(N\mathcal{C}\_{/C})$. Finally, the explicit combinatorics of our category $\mathcal{C}$ and of maps between the $\mathbb{Z}/p^n\mathbb{Z}$ allow us to identify these to be both equivalent to our topos $\mathcal{X}$, and therefore of finite homotopy dimension. Since further $\mathcal{X}$ is generated under colimits by the image of $j$ (note how colimits in the sheaf category, being a reflective subcategory of $\mathcal{P}(N\mathcal{C})$, are constructed), we see by the very definition that $\mathcal{X}$ must also be locally of finite homotopy dimension.


