Sphere spectrum

In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum of S⁰, i.e., a set of two points. Explicitly, the nth space in the sphere spectrum is the n-dimensional sphere Sⁿ, and the structure maps from the suspension of Sⁿ to Sⁿ⁺¹ are the canonical homeomorphisms. The k-th homotopy group of a sphere spectrum is the k-th stable homotopy group of spheres.

The localization of the sphere spectrum at a prime number p is called the local sphere at p and is denoted by ${\displaystyle S_{(p)}}$.