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9 votes
0 answers
467 views

Using higher topos theory to study Cech cohomology

It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ordin …
Markus Zetto's user avatar
9 votes
1 answer
681 views

Coherent objects in a hypercomplete $\infty$-topos

In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\mathcal{X}^{coh …
Markus Zetto's user avatar
2 votes
1 answer
472 views

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 $n$-connectiv …
Markus Zetto's user avatar
2 votes
1 answer
172 views

(Local) Homotopy dimension of $\infty$-topoi on paracompact spaces

I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a continuous map $p: …
Markus Zetto's user avatar
2 votes
1 answer
198 views

Subcategory of coherent objects in an $\infty$-topos forming a local $\infty$-pretopos

My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following: Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the fu …
Markus Zetto's user avatar