comodule spectrum

[[!include stable homotopy theory - contents]]

[[!include higher algebra - contents]]

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 $X$.

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

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

See also at *A-theory*.

- 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 14:35:23. See the history of this page for a list of all contributions to it.