nLab coring spectrum

Idea

A coring spectrum is a comonoid object in the symmetric monoidal (infinity,1)-category of spectra. The concept of a coring spectrum is to that of a ring spectrum like a coalgebra is to an associative algebra.

Examples

Example

(suspension spectra carry canonical structure of coring spectra)
Every $\infty$-groupoid (homotopy type of a topological space) $X$ is canonically a comonoid object in the Cartesian monoidal (infinity,1)-category ∞Grpd (here). Accordingly, since forming suspension spectra is strong monoidal (see there), its suspension spectrum $\Sigma^\infty X_+$ is a coring spectrum (via the smash-monoidal diagonals).

For more on this coring structure on suspension spectra see also (here) at suspension spectrum and see discussion of A-theory as in Hess & Shipley 2014.

Related concepts

• comodule spectrum

References

The canonical coring-spectrum structure on suspension spectra is used in

• Kathryn Hess, Brooke Shipley, Waldhausen K-theory of spaces via comodules, Advances in Mathematics 290 (2016): 1079-1137 [arXiv:1402.4719, doi:10.1016/j.aim.2015.12.019]

(for discussion of A-theory).