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
By $Spectrum$ or $Spectra$ one may denote the stable (∞,1)-category of spectra. It is also denoted $Sp$, or sometimes $Spec$ although that can be confusing.
Its homotopy category is the classical stable homotopy category.
For more see the entry stable (∞,1)-category of spectra.
A sequence of morphisms of spectra $E \longrightarrow F \longrightarrow G$ is a homotopy fiber sequence if and only if it is a homotopy cofiber sequence:
In fact:
A homotopy-commuting square in Spectra is a homotopy pullback if and only it is a homotopy pushout.
This follows from Prop. by the fact that Spectra is additive (this Prop.).
See also arXiv:1906.04773, Prop. 6.2.11, MO:q/132347.
This property of Spectra (Prop. , Prop. ) reflects one of the standard defining axioms on stable (∞,1)-categories (see there) and on stable derivators (see there).
$Spectra$ is equivalently
the free stable locally presentable (∞,1)-category on one compact generator, namely the sphere spectrum.
the stable (infinity,1)-category of quasicoherent infinity-stacks on Spec(S).
Last revised on January 16, 2021 at 14:58:27. See the history of this page for a list of all contributions to it.