Using a construction closely related to Waldhausen's $S_\bullet$-construction, we produce a spectrum $K(\mathbf{ Var}_{ /k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0 (\mathbf{ Var}_{ /k})$. We then produce liftings of various motivic measures to spectrum-level maps, including maps into Waldhausen's $K$-theory of spaces $A(\ast)$ and to $K(\mathbf{ Q})$. We end with a conjecture relating $K(\mathbf{ Var}_{ /k})$ and the doubly-iterated $K$-theory of the sphere spectrum.
Categories: math.AT math.AG math.KT