nLab comodule spectrum

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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.

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]}

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

Last revised on March 7, 2017 at 19:35:23. See the history of this page for a list of all contributions to it.