nLab stable weak homotopy equivalence



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


Equality and Equivalence



The analog in stable homotopy theory of weak homotopy equivalences in classical homotopy theory.

The stable weak equivalences of sequential spectra in simplicial sets form the weak equivalences in the Bousfield-Friedlander model structure for stable homotopy theory.

Beware that for other types of spectra there may be subtle corrections to this statement. For instance for symmetric spectra the maps that are stable weak equivalences on the underlying sequential spectra are guaranteed to be weak equivalences in the model structure on symmetric spectra only on semistable symmetric spectra.


For sequential spectra


The stable homotopy groups of a sequential spectrum XX, is the \mathbb{Z}-graded abelian groups given by the colimit of homotopy groups of the component spaces (or of their geometric realization if they are given as simplicial sets)

π (X)lim kπ +k(X n). \pi_\bullet(X) \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{\bullet+k}({ X_n }) \,.

This constitutes a functor

π :SeqSpec(sSet)Ab . \pi_\bullet \;\colon\; SeqSpec(sSet) \longrightarrow Ab^{\mathbb{Z}} \,.

A morphism f:XYf \colon X \longrightarrow Y of sequential spectra, is called a stable weak homotopy equivalence, if its image under the stable homotopy group-functor of def. is an isomorphism

π (f):π (X)π (Y). \pi_\bullet(f) \;\colon\; \pi_\bullet(X) \longrightarrow \pi_\bullet(Y) \,.


Closure properties

(e.g. MMSS 00, theorem 7.4)


Last revised on May 20, 2016 at 11:18:24. See the history of this page for a list of all contributions to it.