nLab
comodule spectrum

Contents

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

Dual to the concept of a module spectrum over a ring spectrum is a comodule spectrum over a coring spectrum, the analog in stable homotopy theory of the concept of comodules in algebra and homological algebra.

Properties

Over suspension spectra

The suspension spectrum Σ X=𝕊[X]\Sigma^\infty X = \mathbb{S}[X] of any ∞-groupoid (homotopy type of a topological space) XX is canonically a coring spectrum by the fact that every XX, is uniquely a coalgebra object? in the Cartesian monoidal (∞,1)-category ∞Grpd via the diagonal (here), and using that Σ \Sigma^\infty is a strong monoidal functor.

If XX is connected object in an (∞,1)-topos (the homotopy type of a connected topological space) then 𝕊[X]\mathbb{S}[X]-comodule spectra are equivalently module spectra over the ∞-group ∞-ring 𝕊[ΩX]\mathbb{S}[\Omega X] of the loop space ∞-group of XX.

CoModSpectra 𝕊[X]ModSpectra 𝕊[ΩX] CoModSpectra_{\mathbb{S}[X]} \;\simeq\; ModSpectra_{\mathbb{S}[\Omega X]}

(Hess-Shipley 14, theorem 1.2 with prop. 5.18)

See also at A-theory.

References

Last revised on March 7, 2017 at 14:35:23. See the history of this page for a list of all contributions to it.