nLab A-theory

Contents

Context

Representation theory

representation theory

geometric representation theory

Contents

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).

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

and the interpretation in temrs of $\mathbb{S}[\Omega X]$-module spectra and Koszul dually in terms of $\mathbb{S}[X]$-comodule spectra is due to

Last revised on September 4, 2020 at 08:22:05. See the history of this page for a list of all contributions to it.