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 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 $G = 1$.

Definition

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 $W$.

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

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

• If we drop the requirement that the maps $\tilde \sigma_{V,W}$ be homeomorphisms, we obtain the notion of a prespectrum.

Remark

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.

Related concepts

• excisive functor

• functor with smash product

References

• Anthony Elmendorf, Igor Kriz, Peter May; section 1 of: Modern foundations for stable homotopy theory, in: Ioan Mackenzie James (ed.): Handbook of Algebraic Topology, North Holland (1995) 213–253 [pdf]

• Stanley Kochmann; section 3.3 of: Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs 7, AMS (1996) [ams:fim-7]