# 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. ## 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.