nLab
A-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebraic topology

Representation theory

Contents

Idea

Waldhausen’s A-theory (Waldhausen 85) of a connected homotopy type XX is the algebraic K-theory of the suspension spectrum Σ + (ΩX)\Sigma^\infty_+ (\Omega X) of the loop space ΩX\Omega X, hence of the ∞-group ∞-rings 𝕊[ΩX]\mathbb{S}[\Omega X] of the looping ∞-group ΩX\Omega X, hence the K-theory of the parametrized spectra over XX (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 XX is due to

and the interpretation in temrs of 𝕊[ΩX]\mathbb{S}[\Omega X]-module spectra and Koszul dually in terms of 𝕊[X]\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.