For various constructions in stable homotopy theory – such as notably that of the symmetric monoidal smash product of spectra – it is useful to use a model for objects in the stable (∞,1)-category of spectra and the stable homotopy category more refined than that given by sequential spectra. The notion of coordinate-free spectrum is such a refinement.
Where a sequential spectrum is a collection of pointed topological space indexed by the natural numbers , 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 .
In the broader context of equivariant stable homotopy theory a coordinate-free spectrum may be thought of as the special case of a G-spectrum over a G-universe for the special case of the trivial group .
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