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
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
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.
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)
This constitutes a functor
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
(e.g. MMSS 00, theorem 7.4)
Last revised on May 20, 2016 at 15:18:24. See the history of this page for a list of all contributions to it.