nLab stable weak homotopy equivalence

Idea

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.

Definition

For sequential spectra

Definition

The stable homotopy groups of a sequential spectrum $X$, 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)

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

This constitutes a functor

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

Definition

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

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

Properties

Closure properties

(e.g. MMSS 00, theorem 7.4)

References

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)