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Welcome to the Spectral wiki!
For now we'll use the wiki to organize the overall plan of attack, and informal explanations of the ideas involved, links, etc. Specific tasks should be made issues.
We want to formalize some spectral sequences, in particular the Serre spectral sequence.
We get a spectral sequence from an exact couple. Sometimes, we can get an exact couple from a filtered complex, and we even get two of these when we have a double complex.
In HoTT, Mike has described how to get spectral sequences (via exact couples) in two blog posts:
- What is a Spectral Sequence?
- Spectral Sequences
- Spectral sequences in HoTT (this is a consolidated version the above two that includes images missing from the second one.)
Related constructions are described in Rognes' notes on The Adams Spectral Sequence and in VII.7 of Goerss-Jardine's “Simplicial homotopy theory” (later in the “Modern Birkhäuser Classics”-version it became section IV.2).
Spectral sequence can accommodate a lot of algebraic structure. On the one hand, we can start with an exact couple in any abelian category. On the other hand, we have concrete abelian groups with extra structure: grading, module over a ring, algebra structure, etc.
We'll improve this as we learn how things hang together:
Here's a rough sketch of where stand as of 9/2016:
- LES of pi's of a ptd map. (check)
- same for a map of spectra
- exact couple from iterated sequence of maps of spectra
- SS of an exact couple
- iterated map of spectra corresponding to Serre SS.
- Jeremy: homological algebra
- Clive and Steve: Book, Chapter 8
- Ulrik: Hopf fibrations (section 8.5)
- Floris: LES for a fibration
- Egbert: Exact couples
- ? : Spectra and spectrification
- ? (Floris or Ulrik): Truncated 2-quotients (thus Rezk completion, K(G,1), …)
- You: