nLab comodule spectrum

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

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

$$CoModSpectra_{\mathbb{S}[X]} \;\simeq\; ModSpectra_{\mathbb{S}[\Omega X]}$$

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

See also at A-theory.

Related concepts

• dg-comodule

References

• Kathryn Hess, Brooke Shipley, Waldhausen K-theory of spaces via comodules, Advances in Mathematics 290 (2016): 1079-1137 (arXiv:1402.4719)