nLab homotopy spectral sequence



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




The notion of homotopy spectral sequence adapts the notion of spectral sequence from homological algebra/stable homotopy theory to unstable/non-abelian homotopy theory.

Where a spectral sequence consists of objects and morphisms in an abelian category – which typically come from the (co-)homology groups of a (co-)chain complex or generally from the homotopy groups of a stable homotopy type – the notion of homotopy spectral sequence is modeled on the homotopy groups of (pointed) topological spaces/homotopy types which in the lowest two degrees consist instead of nonabelian groups and (pointed) sets.


The classical and motivating examples of a homotopy spectral sequence is the Bousfield-Kan spectral sequence which computes homotopy groups of a topological space/simplicial set/homotopy type realized as the totalization of a cosimplicial homotopy type. It may be regarded as the unstable analog of the Adams spectral sequence, which computes homotopy groups of certain spectra. The corresponding spectral sequence for homology groups of the totalization of a cosimplicial homotopy type is the Eilenberg-Moore spectral sequence.


The Bousfield-Kan spectral sequence was introduced and originally studied in

  • Aldridge Bousfield, Daniel Kan, The homotopy spectral sequence of a space with coefficients in a ring. Topology, 11, pp. 79–106, 1972.

  • Aldridge Bousfield, Daniel Kan, A second quadrant homotopy spectral sequence, Transactions of the American Mathematical Society Vol. 177, Mar., 1973

  • Aldridge Bousfield, Daniel Kan, Pairings and products in the homotopy

    spectral sequence_. Transactions of the American Mathematical Society, 177, pp. 319–343, 1973.

  • Aldridge Bousfield, Homotopy Spectral Sequences and Obstructions, Isr. J. Math. 66 (1989), 54-104.

An early textbook account of this work is in

A general abstract category-theoretic discussion is in

Lecture notes on the Bousfield-Kan spectral sequence include

  • Bertrand Guillo, The Bousfield-Kan spectral sequence pdf

Discussion of this using effective homology includes

  • Romero Ibanez, Effective homology and spectral sequences, 2007 (pdf)

Discussion in homotopy type theory is in

Last revised on November 2, 2014 at 17:38:08. See the history of this page for a list of all contributions to it.