# 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 $$\mathcal{X}=Shv(N\mathcal{C})$$. 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.

• If $B\mathbf{Z}_p$ is of finite homotopy dimension then it also locally of finite homotopy dimension, because every finite index subgroup of $\mathbf{Z}_p$ is isomorphic to $\mathbf{Z}_p$. For an example of something locally but not globally of finite homotopy dimension: the slice $\infty$-category $\mathcal{S}_{/X}$, where $X$ is any space which is not finitely dominated. Apr 2 at 16:45
• Thank you very much for your answer, that does indeed make a lot of sense! Apr 2 at 17:43

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 finitely dominated - the fact that this must have infinite homotopy dimension follows directly from HTT 7.2.4.1, which implies that if it were finite, our space $$X$$ would have to be a retract of a finite 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.