Timeline for How can one concretely approximate a homotopy type by a scheme or (higher/derived) stack?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2018 at 15:10 | comment | added | Jon Pridham | Of course, for any homotopy type $X$ you have an associated $\infty$-stack given by hypersheafifiying the constant presheaf $X$. $K(G,1)$ would then usually be called $BG$. [Etale homotopy types go in the opposite direction, producing homotopy types from stacks.] | |
| Mar 3, 2018 at 11:04 | comment | added | Jon Pridham | The issue with schematic homotopy types comes entirely from π1; for instance the pro-reductive completion of an abelian group is Spec of its character ring (usually huge). Smaller relative completions are more manageable; completion relative to 1 just gives Sullivan's rational homotopy types, regarded as stacks by passing to the associated cosimplicial algebra. | |
| Mar 3, 2018 at 6:41 | comment | added | Tim Porter | Perhaps you should indicate what you mean by 'an arbitrary homotopy type' and also how you would like `'interesting' to be interpreted. Pro-finite homotopy theory is the obvious place to look. | |
| Mar 3, 2018 at 2:07 | comment | added | user40276 | ... that have the $G_F$ as a $\pi^1$ and cohomology on $Z/n$ agreeing with Galois cohomology. Btw, I don't know how étale homotopy theory should be on the pro-étale site instead of the étale. | |
| Mar 3, 2018 at 2:07 | comment | added | user40276 | I'm aware of only one way of attaching homotopy types to algebraic objects. You can take the étale pro homotopy type a derived higher stack arxiv.org/abs/1506.07155 . Étale homotopy types were first defined by Artin-Mazur and refined by Frieddlander. Later Hoyois gave this better description. I don't know though what would be the correct site for derived schemes/stack. In the case of $K (G, 1)$, any field with absolute Galois group G will do the job. There's also Scholze-Kucharczyk paper arxiv.org/pdf/1609.04717.pdf constructing a space functorially for fields of char $0$ ... | |
| Mar 2, 2018 at 23:53 | history | edited | David Handelman | CC BY-SA 3.0 |
spelling, typos, non is not a word
|
| Mar 2, 2018 at 23:46 | review | First posts | |||
| Mar 2, 2018 at 23:53 | |||||
| Mar 2, 2018 at 23:43 | history | asked | Patrick Elliott | CC BY-SA 3.0 |