finite spectrum in nLab

Contents

Contents

Idea

A finite spectrum is a spectrum presented by sequences of finite homotopy types (finite CW-complexes). Precisely, they are the CW-spectra with a finite number of cells of a spectrum.

Properties

Finite spectra are the compact objects in the stable homotopy category. The full subcategory on finite spectra is equivalently the Spanier-Whitehead category on finite CW complexes (e.g. Schwede 12, chapter II, section 7).

Many of the basic constructions and theorems in chromatic homotopy theory apply to finite p-local spectra, such as

• telescopic localization

• periodicity theorem

• chromatic convergence theorem

• Spanier-Whitehead category

• finite set, finite number

• finite topological space, finite CW-complex

• finite type

• p-local spectrum

References

• Neil Strickland, Section 2.2 of An introduction to the category of spectra (pdf)

• Stefan Schwede, chapter II, section 7 of Symmetric spectra, 2012 (pdf)