nLab spectral sequence of a simplicial stable homotopy type

Idea

For $\mathcal{C}$ a stable (∞,1)-category and $X_\bullet$ a simplicial object in an (∞,1)-category in $\mathcal{C}$, then the simplicial skeleta of $X$ give it the structure of a filtered object in an (∞,1)-category. The corresponding spectral sequence of a filtered stable homotopy type has as its first page the Moore complexes of the corresponding simplicial objects of homotopy groups.

Examples

• for the Cech nerve of an E-∞ algebra $E$ this yields the $E$-Adams spectral sequence.

Related concepts

tower diagram/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

References

The general theory is developed in

• Jacob Lurie, section 1.2.4 of Higher Algebra

Quick review is in

• Dylan Wilson section 1.2 of Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra lecture at 2013 Pre-Talbot Seminar (pdf)