Timeline for Linear $\infty$-categories $\mathrm{QC(X)}$ and $\mathrm{Perf(X)}$ of a "derived" stack $\mathrm{X}$
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2017 at 23:14 | history | edited | jereckherr | CC BY-SA 3.0 |
deleted 3 characters in body
|
| Jun 17, 2017 at 21:28 | comment | added | jereckherr | @DavidBen-Zvi Also, the notion of generalized space taken from Lurie's paper I was referring to is that: ncatlab.org/nlab/show/derived+Deligne-Mumford+stack | |
| Jun 17, 2017 at 21:26 | history | edited | jereckherr | CC BY-SA 3.0 |
added 148 characters in body
|
| Jun 17, 2017 at 21:23 | comment | added | jereckherr | @DenisNardin I suspected this is the case. I suppose the dual to my question is "what can derived/spectral stacks be over"? | |
| Jun 17, 2017 at 21:22 | comment | added | jereckherr | @DavidBen-Zvi So, there is a notion of a \infty-category linear over a simplicial commutative ring? I was under the impression that derived Artin stacks are still studied over a commutative ring, as they are built from simplicial k-algebras (k being an ordinary commutative ring). Also, Lurie in his "Structured Spaces" still considers generalized schemes (which includes both derived and spectral stacks) over a ring k (not a generalized ring). | |
| Jun 17, 2017 at 21:21 | comment | added | Denis Nardin | @jereckherr What is $X$ a derived stack over? The answer to this question is also the answer to yours. | |
| Jun 17, 2017 at 21:20 | comment | added | jereckherr | @HughThomas Thank you for the advice. I've made an edit. | |
| Jun 17, 2017 at 21:19 | history | edited | jereckherr | CC BY-SA 3.0 |
added 812 characters in body
|
| Jun 17, 2017 at 20:17 | comment | added | Hugh Thomas | This question will likely attract more attention if it is clearer what you are really asking. "But what about is the base ring of of" comes at a key point and doesn't make grammatical sense, which isn't helping. | |
| Jun 17, 2017 at 20:15 | comment | added | David Ben-Zvi | If X is a derived stack over k, QC(X) is a k-linear stable infinity category. This holds whether k is E_infty or simplicial commutative or ordinary discrete commutative (in which case you may prefer to think of it as a k linear dg category). Same for Perf. geometrically this is just expressing the structure map from X to Spec k. | |
| Jun 17, 2017 at 19:56 | comment | added | jereckherr | @DylanWilson What type of rings QC(X) is linear over? Intuition may say that over E_infinity-rings, but if we X is a derived Artin stack, apparently, QC(X) is not linear over a simplicial commutative ring, but over an ordinary commutative ring. What is more, Lurie in his "Structures Spaces" define generalized schemes (a general theory that includes ordinary schemes, algebraic stacks, derived Artin stacks, derived schemes, spectral schemes, spectral Deligne-Mumford stacks et. al) over a commutative ring k. Though he doesn't develop sheaf theory for such generalized schemes. | |
| Jun 17, 2017 at 18:55 | comment | added | Dylan Wilson | What is the question exactly? Whether QC(X) is linear over some E_infty-ring? A stupid answer is: of course, the sphere spectrum. | |
| Jun 17, 2017 at 18:25 | review | First posts | |||
| Jun 17, 2017 at 20:18 | |||||
| Jun 17, 2017 at 18:20 | history | asked | jereckherr | CC BY-SA 3.0 |