Weight Filtrations and Derived Motivic Measures

Let $k$ be a field admitting resolution of singularities. We lift a number of motivic measures, such as the Gillet-Soul\'e measure and the compactly supported $\mathbb{A}^1$-Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soul\'e weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a $k$-variety $X$, the collection of all Gillet-Soul\'e weight complexes of $X$ form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the $K$-theory of a Waldhausen category is equivalent to the $K$-theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soul\'e weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over $k$. We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of $K(Var_k)$, the Zakharevich $K$-theory of varieties.