The determination of quadratic forms throughout sheaves A and Ω, and functorial morphisms, play an important role in the construction of some categories of quadratic differential triads over X, 〖id〗_X,〖id〗_(X × 〖id〗_X ),… In this paper, we generalize these notions so that we obtain categories of quadratic differential triads over the categories 〖Open〗_X and TOP, respectively. We also treat the notions of quadratic integral triads. The cohomology, homology and resolution notions are introduced with some applications in Electromagnetism.