62 lines
3.6 KiB
Markdown
62 lines
3.6 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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- Mac Lane's Homology contains a lot of homological algebra and a chapter on spectral sequences, including exact couples.
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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](http://ncatlab.org/nlab/show/module), [vector spaces](http://ncatlab.org/nlab/show/vector+space),
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- some basic theory: product, tensor, hom, projective,
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- categories of algebras, [abelian categories](http://ncatlab.org/nlab/show/abelian+category),
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- exact sequences, short and long
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- [snake lemma](http://ncatlab.org/nlab/show/snake+lemma)
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- [5-lemma](http://ncatlab.org/nlab/show/five+lemma)
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- [chain complexes](http://ncatlab.org/nlab/show/chain+complex) and [homology](http://ncatlab.org/nlab/show/homology)
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- [exact couples](http://ncatlab.org/nlab/show/exact+couple), probably just of Z-graded objects, and derived exact couples
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- spectral sequence of an exact couple
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- [convergence of spectral sequences](http://ncatlab.org/nlab/show/spectral+sequence#ConvergenceOfSpectralSequences)
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### Topology To Do:
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- fiber sequence
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+ already have the LES
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+ need shift isomorphism
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+ Hom'ing into a fiber sequence gives another fiber sequence.
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- cofiber sequences
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+ Hom'ing out gives a fiber sequence: if `A → B → coker f` cofiber
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sequences, then `X^A → X^B → X^(coker f)` is a fiber sequence.
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- [prespectra](http://ncatlab.org/nlab/show/spectrum+object) and [spectra](http://ncatlab.org/nlab/show/spectrum), suspension
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+ try indexing on arbitrary successor structure
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+ think about equivariant spectra indexed by representations of `G`
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- [spectrification](http://ncatlab.org/nlab/show/higher+inductive+type#spectrification)
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+ adjoint to forgetful
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+ as sequential colimit, prove induction principle (if useful)
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+ connective spectrum: `is_conn n.-2 Eₙ`
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- [parametrized spectra](http://ncatlab.org/nlab/show/parametrized+spectrum), parametrized smash and hom between types and spectra
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- fiber and cofiber sequences of spectra, stability
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+ limits are levelwise
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+ colimits need to be spectrified
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- long exact sequences from (co)fiber sequences of spectra
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+ indexed on ℤ, need to splice together LES's
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- Postnikov towers of spectra
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+ basic definition already there
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+ fibers of Postnikov sequence unstably and stably
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- exact couple of a tower of spectra
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+ need to splice together LES's
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### Already Done:
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- Most things in the HoTT Book up to Section 8.9 (see [this file](https://github.com/leanprover/lean/blob/master/hott/book.md))
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- pointed types, maps, homotopies and equivalences
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- definition of algebraic structures such as groups, rings, fields
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- some algebra: quotient, product, free groups.
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- [Eilenberg-MacLane spaces](http://ncatlab.org/nlab/show/Eilenberg-Mac+Lane+space) and EM-spectrum
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