The Comprehensive Factorization and Torsors

This is an expanded, revised and corrected version of the first author’s preprint [1]. The discussion of one-dimensional cohomology H in a fairly general category E involves passing to the (2-)category Cat(E) of categories in E . In particular, the coefficient object is a category B in E and the torsors that Hclassifies are particular functors in E . We only impose conditions on E that are satisfied also by Cat(E) and argue that H for Cat(E) is a kind of H for E , and so on recursively. For us, it is too much to ask E to be a topos (or even internally complete) since, even if E is, Cat(E) is not. With this motivation, we are led to examine morphisms in E which act as internal families and to internalize the comprehensive factorization of functors into a final functor followed by a discrete fibration. We define B-torsors for a category B in E and prove clutching and classification theorems. The former theorem clutches Čech cocycles to construct torsors while the latter constructs a coefficient category to classify structures locally isomorphic to members of a given internal family of structures. We conclude with applications to examples.