1.1. QCoh on a stack. We know that QCoh forms a stack, i.e., sheaf of groupoids, over Schfpqc(S) for any scheme S. Thus if we have an fpqc sheaf of groupoids X over S, we can define QCoh(X ) as maps of sheaves X → QCoh on Schfpqc(S). By 2-Yoneda, this definition agrees with the usual notion of quasicoherent sheaves if X is a scheme. For various other (usually equivalent) definitions in the case...