In this talk I will discuss a new construction of the K-theory spectrum of varieties K(Var), using the framework of so-called squares categories. With this approach, a spectral analogue of Bittner's presentation can be formulated (but remains conjectural): is K(Var) equivalent to a spectrum built from just smooth and complete varieties?
In the second half of the talk, I define a category of "compacly supported cohomology theories". Here squares show up again, and one can show that a compactly supported cohomology theory is determined by its image on smooth and complete varieties, which could remind one of Bittner's presentation. A connection between K(Var) and this notion of compactly supported cohomology theories exists, in the form of a derived motivic measure. This talk is based on joint work with Jonathan Campbell, Mona Merling and Inna Zakharevich.