nLab A-theory

Idea

Waldhausen’s A-theory (Waldhausen 1985) of a connected homotopy type $X$ is the algebraic K-theory of the suspension spectrum $\Sigma^\infty_+ (\Omega X)$ of the loop space $\Omega X$, hence of the ∞-group ∞-ring $\mathbb{S}[\Omega X]$ of the looping ∞-group $\Omega X$, hence is the K-theory of the parametrized spectra over $X$ (Hess & Shipley 2014).

Related concepts

• iterated algebraic K-theory

• coring spectrum

References

The original definition

• Friedhelm Waldhausen: Algebraic K-theory of spaces, in: Algebraic and Geometric Topology (Proceedings of a conference at Rutgers, New Brunswick, N. J., 1983), Lecture Notes in Math. 1126, Springer (1985) 318-419 [doi:10.1007/BFb0074449, pdf, pdf]

The interpretation in terms of certain module spectra over the Spanier-Whitehead dual of $X$ is due to

• Andrew Blumberg, Michael Mandell, Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces, J. Topology 4 2 (2011) 327-342 [arXiv:0912.1670, doi:10.1112/jtopol/jtr003]

and the interpretation in terms of $\mathbb{S}[\Omega X]$-module spectra, and Koszul dually in terms of $\mathbb{S}[X]$-comodule spectra (using the canonical coring spectrum-structure of suspension spectra), is due to:

• 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]