| Title:
|
On the étale homotopy type of higher stacks (English) |
| Author:
|
Carchedi, David |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
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2209-0606 |
| Volume:
|
5 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
121-185 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A new approach to étale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the big étale site, and in particular to all algebraic stacks. This approach also produces a more refined invariant than the original construction of Artin-Mazur [2], namely we produce a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\Cal X$ is an arbitrary higher stack on the étale site of affine schemes of finite type over $\Bbb C$, then the étale homotopy type of $\Cal X$ agrees with the homotopy type of the underlying stack $\Cal X_{top}$ on the topological site, after profinite completion. In particular, if $\Cal X$ is an Artin stack locally of finite type over $\Bbb C$, our definition of the étale homotopy type of $\Cal X$ agrees up to profinite completion with the homotopy type of the underlying topological stack $\Cal X_{top}$ of $\Cal X$ in the sense of Noohi [35]. We also show this comparison is compatible in a suitable sense with the comparison theorem of Friedlander for simplicial schemes [17]. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of $\infty$-categories which we believe to be of independent interest. (English) |
| Keyword:
|
étale homotopy |
| Keyword:
|
higher categories |
| Keyword:
|
stacks |
| Keyword:
|
topos theory |
| MSC:
|
14A20 |
| MSC:
|
14F35 |
| MSC:
|
18B25 |
| MSC:
|
55N25 |
| idZBL:
|
Zbl 1493.14033 |
| idMR:
|
MR4367219 |
| DOI:
|
10.21136/HS.2021.04 |
| . |
| Date available:
|
2026-03-13T05:32:48Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153436 |
| . |
| Reference:
|
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