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