# 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 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.