Timeline for Why are spectral sequences so ubiquitous?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jan 7, 2010 at 17:08 | comment | added | Clark Barwick | Tyler is of course right. Perhaps one should say instead that all "natural" spectral sequences come from filtered objects in an ∞-category $C$ with a conservative functor $C\to D$ to a stable ∞-category with a t-structure. The terms of the spectral sequence live in the heart of the t-structure, and if $C$ isn't already stable, then you get "fringe effects," which in general can be a lot more disastrous than the gentle term suggests. This perspective doesn't explain the presence of non-abelian groups and pointed sets in, e.g., the spectral sequence of a tower of fibrations, though. | |
| Dec 16, 2009 at 13:09 | comment | added | Tyler Lawson | With respect to your edit, there are several natural spectral sequences abutting to homotopy groups of certain spaces, which are less likely to arise from filtered spectra. For example, the unstable Adams spectral sequence for computing homotopy from cohomology rings. | |
| Dec 16, 2009 at 3:59 | history | edited | Reid Barton | CC BY-SA 2.5 |
request for counterexamples
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| Dec 8, 2009 at 3:12 | comment | added | Reid Barton | Oops. There are various ways a spectral sequence can yield a long exact sequence, but you're definitely right about the one I care about. | |
| Dec 8, 2009 at 3:10 | history | edited | Reid Barton | CC BY-SA 2.5 |
deleted 19 characters in body
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| Dec 8, 2009 at 2:28 | comment | added | VA. | Just a typo correction: the long exact sequence is formed by the entries in the $E_1$ page and the limit of the spectral sequence, not $E_2$ (or $E^1$ and not $E^2$ if you prefer homological spectral sequences to the cohomological ones). | |
| Dec 7, 2009 at 19:08 | history | answered | Reid Barton | CC BY-SA 2.5 |