nLab Spectrum



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


Stable Homotopy theory



By SpectrumSpectrum or SpectraSpectra one may denote the stable (∞,1)-category of spectra. It is also denoted SpSp, or sometimes SpecSpec although that can be confusing.

Its homotopy category is the classical stable homotopy category.

For more see the entry stable (∞,1)-category of spectra.


Finite homotopy (co)limits of spectra


A sequence of morphisms of spectra EFGE \longrightarrow F \longrightarrow G is a homotopy fiber sequence if and only if it is a homotopy cofiber sequence:

A proof is spelled out at Introduction to Stable homotopy theory (this Prop., following Lewis-May-Steinberger 86, chapter III, theorem 2.4 )

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).

Other abstract characterizations

SpectraSpectra is equivalently

category: category

Last revised on January 16, 2021 at 14:58:27. See the history of this page for a list of all contributions to it.