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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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
Discussion of this using effective homology includes
Discussion in homotopy type theory is in
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