nLab
coordinate-free spectrum

Idea

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 $\Omega$-spectra. The notion of coordinate-free spectrum is such a refinement.

Where an $\Omega$-spectrum is a collection of topological spaces indexed by the integers $\mathbb{Z}$, 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 ${ℝ}^{\mathrm{\infty }}$.

Definition

Let $U$ be a real inner product vector space isomorphic to the direct sum ${ℝ}^{\mathrm{\infty }}$ of countably many copies of the real line $ℝ$.

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 ${\Omega }^{V}X:=\mathrm{Maps}(S^V,X)$ 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 $W$.

Definition

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

  $${\tilde{\sigma }}_{V,W}:E_V\stackrel{\simeq }{\to }{\Omega }^{W-V}{E}_W.$$\tilde \sigma_{V,W} : E_V \stackrel{\simeq}{\to} \Omega^{W-V} E_W \,.

References

• A. Elmendorf, I. Kriz, P. May, Modern foundations for stable homotopy theory (pdf)