A category of correspondences based on Waldhausen A-theory has interesting
analogies, in the context of differential topology, to categories of mixed Tate
motives studied in arithmetic geometry.

In particular, the Hopf object S \wedge_A S (regarding A(*) as a kind of
local ring over the sphere spectrum) has some similarities to a motivic group
for this category; its associated rational Lie algebra is free, on odd-degree
generators...