Algebraic $K$-theory for squares categories

In this paper we introduce a new formalism for $K$-theory, called squares $K$-theory. This formalism allows us to simultaneously generalize the usual three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B \leftarrow C$, by defining $K_0$ of a squares category to satisfy a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with these corners. Examples that rely on this formalism are $K$-theory of smooth manifolds of a fixed dimension and $K$-theory of (smooth and) complete varieties. Another application we give of this theory is the construction of a derived motivic measure taking value in the $K$-theory of homotopy sheaves.