Fpqc Descent and Grothendieck Topologies in a Differential Setting

1.1. Sites. We begin with the underlying idea for a Grothendieck topology. Consider your favorite topological space X. Its topology, don’t use the Zariski topology, is determined by the partially ordered category UX consisting of all the open subsets of X ordered with respect to inclusion. In order to verify that a presheaf is a sheaf on X, we must first construct all possible coverings U = (Ui ⊂ U)i∈I } of open sets U in X and then verify the two sheaf conditions hold for each covering (Ui ⊂ U)i∈I . But we also want to compare coverings in the following two ways. If (Ui)i∈I is a covering of U and for each i, ( V i j ) j∈Ji is a covering of Ui, then ( V i j ) ∪Ji is also a covering of U. If (Ui)i∈I is a covering of U and V ⊂ U is an open set, then (Ui ∩ V )i∈I is a covering of V. We are just stating that refinements of each open in a covering yields a covering and intersecting a covering with a smaller open set still yields a covering. Keep this example in mind as we go through some category theoretical definitions. Also remember that Ui ∩ Uj = Ui ×X Uj . We will only consider ‘local’ topologies. Let C/X be a category with finite products, i.e. a terminal object X and products of pairs of objects.