Counterexamples concerning $\infty$-topoi with infinite homotopy dimension

Question

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.

Answer 1

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, 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.