nLab A-theory

Idea

Waldhausen’s A-theory (Waldhausen 85) 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 ∞-rings $\mathbb{S}[\Omega X]$ of the looping ∞-group $\Omega X$, hence the K-theory of the parametrized spectra over $X$ (Hess-Shipley 14).

Related concepts

• iterated algebraic K-theory

• coring spectrum

References

The definition is originally due to

• Friedhelm Waldhausen, Algebraic K-theory of spaces Algebraic and geometric topology (Ne Brunswick, N. J., 1983), 318–419, Lecture Notes in Math., 1126, Springer, Berlin, 1985

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 (2011), no. 2, 327-342 (arXiv:0912.1670)

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]