nLab
coordinate-free spectrum

Contents

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 mahbbN\mahbb{N}, a coordinate free spectrum is a collection of topological spaces index by all finite dimensional subspaces of a real inner product vector space UU 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=1G = 1.

Definition

Let UU 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 VUV \subset U a finite-dimensional subspace, write S VS^V for its one-point compactification (an nn-dimensional sphere if VV is nn-dimensional) and for XX any based topological space write Ω VX:=Maps(S V,X)\Omega^V X := Maps(S^V,X) for the topological space of basepoint-preserving continuous maps.

For VWV \subset W an inclusion of finite dimensional subspaces V,WUV,W \subset U write WVW-V for the orthogonal complement of VV in WW.

Definition

A coordinate-free spectrum EE modeled on the “universe” UU is

  • for each finite-dimensional subspace VUV \subset U a pointed topological space E VE_V;

  • for each inclusion VWV \subset W of finite dimensional subspaces V,WUV,W \subset U a homeomorphism of pointed topological spaces

    σ˜ V,W:E VΩ WVE W. \tilde \sigma_{V,W} : E_V \stackrel{\simeq}{\to} \Omega^{W-V} E_W \,.
  • If we drop the requirement that the maps σ˜ V,W\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.

References

Revised on May 2, 2016 14:54:32 by Urs Schreiber (131.220.184.222)