44 lines
2.8 KiB
Markdown
44 lines
2.8 KiB
Markdown
# Spectral Sequences
|
|
|
|
Formalization project of the CMU HoTT group towards formalizing the Serre spectral sequence.
|
|
|
|
#### Participants
|
|
Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Floris van Doorn, Clive Newstead, Egbert Rijke, Mike Shulman.
|
|
|
|
## Resources
|
|
- [Mike's blog post](http://homotopytypetheory.org/2013/08/08/spectral-sequences/) at the HoTT blog.
|
|
- [Mike's blog post](https://golem.ph.utexas.edu/category/2013/08/what_is_a_spectral_sequence.html) at the n-category café.
|
|
- The [Licata-Finster article](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf) about Eilenberg-Mac Lane spaces.
|
|
- We learned about the Serre spectral sequence from [Hatcher's chapter about spectral sequences](https://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html).
|
|
- 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.
|
|
- Mac Lane's Homology contains a lot of homological algebra and a chapter on spectral sequences, including exact couples.
|
|
|
|
## Things to do for Lean spectral sequences project
|
|
|
|
### Algebra To Do:
|
|
- [R-modules](http://ncatlab.org/nlab/show/module), [vector spaces](http://ncatlab.org/nlab/show/vector+space),
|
|
- some basic theory: product, tensor, hom, projective,
|
|
- categories of algebras, [abelian categories](http://ncatlab.org/nlab/show/abelian+category),
|
|
- exact sequences, short and long
|
|
- [snake lemma](http://ncatlab.org/nlab/show/snake+lemma)
|
|
- [5-lemma](http://ncatlab.org/nlab/show/five+lemma)
|
|
- [chain complexes](http://ncatlab.org/nlab/show/chain+complex) and [homology](http://ncatlab.org/nlab/show/homology)
|
|
- [exact couples](http://ncatlab.org/nlab/show/exact+couple), probably just of Z-graded objects, and derived exact couples
|
|
- spectral sequence of an exact couple
|
|
- [convergence of spectral sequences](http://ncatlab.org/nlab/show/spectral+sequence#ConvergenceOfSpectralSequences)
|
|
|
|
### Topology To Do:
|
|
- fiber and cofiber sequences (is this in the library already?)
|
|
- [prespectra](http://ncatlab.org/nlab/show/spectrum+object) and [spectra](http://ncatlab.org/nlab/show/spectrum), suspension
|
|
- [spectrification](http://ncatlab.org/nlab/show/higher+inductive+type#spectrification)
|
|
- [parametrized spectra](http://ncatlab.org/nlab/show/parametrized+spectrum), 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](http://ncatlab.org/nlab/show/Eilenberg-Mac+Lane+space) and spectra
|
|
- Postnikov towers of spectra
|
|
- exact couple of a tower of spectra
|
|
|
|
### Already Done:
|
|
- pointed types
|
|
- definition of algebraic structures such as groups, rings, fields,
|
|
- some algebra: quotient, product, free.
|