Spectral/README.md

# 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:
- HoTT Book sections 8.7, 8.8.
- cofiber sequences
- [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:
- Most things in the HoTT Book up to Section 8.6 (see [this file](https://github.com/leanprover/lean/blob/master/hott/book.md))
- pointed types, maps, homotopies and equivalences
- definition of algebraic structures such as groups, rings, fields
- some algebra: quotient, product, free groups.
Update README.md 2015-12-04 00:59:03 +00:00			`# Spectral Sequences`
more information README.md 2015-11-18 22:09:18 +00:00
update README 2016-03-06 18:23:30 +00:00			`Formalization project of the CMU HoTT group towards formalizing the Serre spectral sequence.`
more information README.md 2015-11-18 22:09:18 +00:00
update README 2016-03-06 18:23:30 +00:00			`#### Participants`
Update README.md 2015-12-04 21:57:32 +00:00			`Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Floris van Doorn, Clive Newstead, Egbert Rijke, Mike Shulman.`
Update README.md 2015-12-04 00:59:03 +00:00
Update README.md 2015-12-04 16:57:56 +00:00			`## Resources`
Update README.md 2015-12-04 21:53:35 +00:00			`- [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.`
Update README.md 2015-12-04 22:00:38 +00:00			`- The [Licata-Finster article](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf) about Eilenberg-Mac Lane spaces.`
Update README.md 2015-12-04 16:57:56 +00:00			`- 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.`
add Mac Lane's Homology to README.md 2015-12-07 08:20:54 +00:00			`- Mac Lane's Homology contains a lot of homological algebra and a chapter on spectral sequences, including exact couples.`
Update README.md 2015-12-04 00:55:16 +00:00
Update README.md 2015-12-04 16:57:56 +00:00			`## Things to do for Lean spectral sequences project`
Update README.md 2015-12-04 00:55:16 +00:00
Update README.md with roadmap based on Mike's blog post 2015-12-04 17:25:18 +00:00			`### Algebra To Do:`
Update README.md 2015-12-06 22:38:55 +00:00			`- [R-modules](http://ncatlab.org/nlab/show/module), [vector spaces](http://ncatlab.org/nlab/show/vector+space),`
Update README.md 2015-12-04 00:55:16 +00:00			`- some basic theory: product, tensor, hom, projective,`
Update README.md 2015-12-06 22:38:55 +00:00			`- categories of algebras, [abelian categories](http://ncatlab.org/nlab/show/abelian+category),`
Update README.md with roadmap based on Mike's blog post 2015-12-04 17:25:18 +00:00			`- exact sequences, short and long`
update README 2016-03-06 18:23:30 +00:00			`- [snake lemma](http://ncatlab.org/nlab/show/snake+lemma)`
Update README.md 2015-12-06 22:38:55 +00:00			`- [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`
Update README.md with roadmap based on Mike's blog post 2015-12-04 17:25:18 +00:00			`- spectral sequence of an exact couple`
Update README.md 2015-12-06 22:38:55 +00:00			`- [convergence of spectral sequences](http://ncatlab.org/nlab/show/spectral+sequence#ConvergenceOfSpectralSequences)`
Update README.md 2015-12-04 00:55:16 +00:00
Update README.md with roadmap based on Mike's blog post 2015-12-04 17:25:18 +00:00			`### Topology To Do:`
update README 2016-03-06 18:23:30 +00:00			`- HoTT Book sections 8.7, 8.8.`
			`- cofiber sequences`
Update README.md 2015-12-06 23:00:19 +00:00			`- [prespectra](http://ncatlab.org/nlab/show/spectrum+object) and [spectra](http://ncatlab.org/nlab/show/spectrum), suspension`
Update README.md 2015-12-06 22:38:55 +00:00			`- [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`
Update README.md with roadmap based on Mike's blog post 2015-12-04 17:25:18 +00:00			`- fiber and cofiber sequences of spectra, stability`
			`- long exact sequences from (co)fiber sequences of spectra`
Update README.md 2015-12-06 22:38:55 +00:00			`- [Eilenberg-MacLane spaces](http://ncatlab.org/nlab/show/Eilenberg-Mac+Lane+space) and spectra`
Update README.md with roadmap based on Mike's blog post 2015-12-04 17:25:18 +00:00			`- Postnikov towers of spectra`
			`- exact couple of a tower of spectra`

			`### Already Done:`
update README 2016-03-06 18:23:30 +00:00			`- Most things in the HoTT Book up to Section 8.6 (see [this file](https://github.com/leanprover/lean/blob/master/hott/book.md))`
			`- pointed types, maps, homotopies and equivalences`
			`- definition of algebraic structures such as groups, rings, fields`
			`- some algebra: quotient, product, free groups.`