For various constructions in stable homotopy theory – such as notably that of the symmetric monoidal smash product of spectra – it is necessary to use a model for objects in the stable (∞,1)-category of spectra and the stable homotopy category more refined than that given by -spectra. The notion of coordinate-free spectrum is such a refinement.
Where an -spectrum is a collection of topological spaces indexed by the integers , a coordinate free spectrum is a collection of topological spaces index by all finite dimensional subspaces of a real inner product vector space isomorphic to .
For a finite-dimensional subspace, write for its one-point compactification (an -dimensional sphere if is -dimensional) and for any based topological space write for the topological space of basepoint-preserving continuous maps.
For an inclusion of finite dimensional subspaces write for the orthogonal complement of in .
A coordinate-free spectrum modeled on the “universe” is