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 $\mathbb{N}$, a coordinate free spectrum is a collection of topological spaces index by all finite dimensional subspaces of a real inner product vector space $U$ isomorphic to $\mathbb{R}^\infty$.
Notice that also spectra realized as excisive functors are coordinate-free in an evident sense, as these are indexed on all finite homotopy types.
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 $G = 1$.
Let $U$ be a real inner product vector space isomorphic to the direct sum $\mathbb{R}^\infty$ of countably many copies of the real line $\mathbb{R}$.
For $V \subset U$ a finite-dimensional subspace, write $S^V$ for its one-point compactification (an $n$-dimensional sphere if $V$ is $n$-dimensional) and for $X$ any based topological space write $\Omega^V X := Maps(S^V,X)$ for the topological space of basepoint-preserving continuous maps.
For $V \subset W$ an inclusion of finite dimensional subspaces $V,W \subset U$ write $W-V$ for the orthogonal complement of $V$ in $W$.
A coordinate-free spectrum $E$ modeled on the “universe” $U$ is
for each finite-dimensional subspace $V \subset U$ a pointed topological space $E_V$;
for each inclusion $V \subset W$ of finite dimensional subspaces $V,W \subset U$ a homeomorphism of pointed topological spaces
If we drop the requirement that the maps $\tilde \sigma_{V,W}$ be homeomorphisms, we obtain the notion of a prespectrum.
The definition of coordinate free spectrum directly generalizes to that of genuine G-spectrum modeled on a G-universe, leading from stable homotopy theory to equivariant stable homotopy theory.
Anthony Elmendorf, Igor Kriz, P. May, section 1 of Modern foundations for stable homotopy theory, in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology (1995) (pdf)
Stanley Kochmann, section 3.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Last revised on September 25, 2016 at 22:23:25. See the history of this page for a list of all contributions to it.