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Fixed a couple incorrect statements at hypercomplete (infinity,1)-topos:
Here’s a simple counterexample to “finite homotopy dimension implies hypercompleteness”:
Let $\mathbf{H}$ be the big (∞,1)-topos on the site $Top$ of topological spaces. Then $\mathbf{H}$ has homotopy dimension $0$ but is not hypercomplete because it contains non-hypercomplete (∞,1)-topoi as essential retracts of its slices. More precisely, if $X$ is a topological space, the embedding $i\colon Open(X)\to Top_{/X}$ is continuous and cocontinuous, so that we have an adjunction $i_!\vdash i^*\vdash i_*$ with $i_!$ fully faithful. In that situation, $i^*$ preserves $n$-truncated morphisms, hence $i_!$ preserves $\infty$-connective morphisms, hence $i^*$ preserves hypercomplete objects, hence $i_!$ preserves non-hypercomplete objects. Thus, for appropriate $X$, $\mathbf{H}_{/X}$ and hence $\mathbf{H}$ are not hypercomplete.
Remark 1 claimed that having enough points in the 1-topos sense implies having enough points in the ∞-sense. I replaced it with a counterexample from HTT.
Thanks for looking into this. But wait, the argument that you removed had just another “hypercomplete” implicit/missing.
The argument said that in the Jardine model structure it is true that when the underlying 1-topos has enough points, then the Jardine weak equivalences are equivalently the stalkwise weak equivalences. Since the Jardine model structure presents the hypercomplete $\infty$-topos over the site, it does follow that this has enough points when the 1-topos does.
I added the remark that the hypercompletion has enough points.
Okay, thanks.
(yeah, that darn hypercompletion subtlety… ;-)
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