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 S<sup>0</sup>, i.e., a set of two points. Explicitly, the nth space in the sphere spectrum is the n-dimensional sphere S<sup>n</sup>, and the structure maps from the suspension of S<sup>n</sup> to S<sup>n+1</sup> 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 .