This thesis consists of four papers, which all contain a certain amount of squares. In Paper I, we study compactly supported cohomology theories of varieties. These can be seen as functors with a nice descent property, out of a category whose objects are varieties, and whose morphisms are spans that consist of an open immersion and a proper map. Using the theory of cd-structures, which are sets of commutative squares that generate a topology, we show that a compactly supported cohomology theory can be uniquely extended from its restriction to smooth and complete varieties. 

In Paper II, we continue the study of cd-structures. If a morphism f between sites satisfies the conditions of the comparison lemma, then it induces an equivalence between the associated categories of (hyper)sheaves. If f is strong symmetric monoidal and the topologies in question are generated by sufficiently nice cd-structures, then we show that f also induces an equivalence between the associated categories of symmetric monoidal hypersheaves. We use this to prove a variant of the main result of Paper I for symmetric monoidal hypersheaves.

The highest degree of square-ness is reached in Paper III. Here, commutative squares are used to build K-theory spectra, most notably for the category of varieties. We reuse some of the square-y arguments from Paper I to show that the K-theory spectrum of the category of varieties is equivalent to the K-theory spectrum of the category of complete varieties. Moreover, exploiting the square-ness of the category of compactly supported cohomology theories that is demonstrated in Paper I, we can construct a new derived motivic measure. Paper III is joint with Jonathan Campbell, Mona Merling and Inna Zakharevich.

In Paper IV, we build on the result of Paper I, and its variation proven in Paper II, to obtain a result about functors that encode six-functor formalisms. Squares show up again, not only in the form of cd-structures, but also in the form of adjointable squares, which play an important role in extending a six-functor formalism from the domain of complete varieties to all varieties.