Talk:Derived algebraic geometry

Derived categories
This section is misleading:

The term "derived" is used in the same way as derived category. It is classic that many operations in algebraic geometry make sense only in the derived category of say quasi-coherent sheaves, rather than the category of such. In the much same way, one usually talks about the ∞-category of derived schemes, etc., as opposed to ordinary category.

It's misleading to say that operations in algebraic geometry only make sense in the derived category. What operations are these? Derived functors... so derived categories are used to make sense of derived functors. So really the question is about deriving functors. So, the "derived" refers to the fact that one is replacing the category of rings with a "derived" category of rings where one has derived operations on rings. The category isn't really the important thing here.

The connection between infinity categories and derived categories is also not very clear here. The point is that derived categories have an infinity categorical enrichment, in fact they are stable infinity categories. Derived rings form an infinity category that is not stable. Analogue: module categories are abelian, category of rings is not. I'll try to get references later but I'll edit it for now.

DrIdiot (talk) 02:54, 27 January 2020 (UTC)

Spectral algebraic geometry
I wonder what is the best way to mention the term "spectral algebraic geometry". The problem is of course that the terms like DAG or SAG do not have the fixed universal meaning; e.g., for some authors, DAG and SAG mean (essentially) the same thing. Instead of pretending there is a single unique approach to this subject, we probably should list several known approaches, one among the should be Lurie's (and that perhaps the best place to mention SAG; and did I just answer my question myself??) —- Taku (talk) 04:30, 11 February 2020 (UTC)
 * Sounds good to me haha — MarkH21talk 19:41, 11 February 2020 (UTC)

Higher Stacks
Simpson's higher stacks should include the axioms for smooth and geometric, and include the lead up to the equivalence over characteristic 0 to "nice enough" higher stacks and differential graded lie algberas concentraded in bounded degrees. This is a super helpful characterization of higher stacks since it makes them more tractable. Also, https://arxiv.org/abs/1105.4853 is immensely helpful in illuminating what a derived stack is using the simplicial language.

Here are some sections which could be included to motivate the study of derived algebraic geometry
 * formal moduli problems, check out https://www.maths.ed.ac.uk/~jpridham/ddt1.pdf https://www.maths.ed.ac.uk/~jpridham/higher.pdf
 * derived deformations/deformations as dg-lie algebras, for complex manifolds check out these two articles by Manetti https://arxiv.org/pdf/math/0507284.pdf https://arxiv.org/pdf/math/0507286.pdf
 * https://perso.math.univ-toulouse.fr/btoen/files/2012/04/dag-ems.pdf is a survey article with more applications, such as deformation quantization and derived intersection theory
 * Here's a link to a recorded school on DAG https://av.tib.eu/media/35240
 * https://arxiv.org/pdf/0804.1274.pdf

Models of Derived Geometry

 * https://arxiv.org/abs/math/0210407
 * https://arxiv.org/abs/math/0212225 — Preceding unsigned comment added by Wundzer (talk • contribs) 05:38, 28 February 2020 (UTC)
 * http://people.math.harvard.edu/~lurie/papers/DAG-VII.pdf
 * https://arxiv.org/abs/math/0212225
 * http://people.maths.ox.ac.uk/~joyce/dmanifolds.html
 * Higher stacks as simplicial schemes https://arxiv.org/abs/0905.4044
 * Deformations of diagrams of commutative algebras https://arxiv.org/abs/1902.10436 (has great dgas controlling the deformation problem) — Preceding unsigned comment added by Wundzer (talk • contribs) 01:08, 12 February 2020 (UTC)