Spectral/README.md

Spectral Sequences

Formalization project of the CMU HoTT group towards formalizing the Serre spectral sequence.

Currently in the project: Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Floris van Doorn, Egbert Rijke, Mike Shulman.

Resources

• Mike's blog post
• We learned about the Serre spectral sequence from Hatcher's chapter about spectral sequences.
• Lang's algebra (revised 3rd edition) contains a chapter on general homology theory, with a section on spectral sequences. Thus, we can use this book at least as an outline for the algebraic part of the project.

Things to do for Lean spectral sequences project

Algebra To Do:

• R-modules, vector spaces,
• some basic theory: product, tensor, hom, projective,
• categories of algebras, abelian categories,
• exact sequences, short and long
• snake lemma (Jeremy)
• 5-lemma
• chain complexes and homology
• exact couples, probably just of Z-graded objects
• derived exact couples
• spectral sequence of an exact couple
• convergence of spectral sequences

Topology To Do:

• pointed types, fiber and cofiber sequences (is this in the library already?)
• prespectra and spectra, suspension
• spectrification
• parametrized smash and hom between types and spectra
• fiber and cofiber sequences of spectra, stability
• long exact sequences from (co)fiber sequences of spectra
• Eilenberg-MacLane spaces and spectra
• Postnikov towers of spectra
• exact couple of a tower of spectra

Already Done:

• definition of algebraic structures such as groups, rings, fields,
• some algebra: quotient, product, free.