This is an expanded, revised and corrected version of the first author's 1981 preprint. The discussion of one-dimensional cohomology H¹ in a fairly general category ℇ involves passing to the 2-category Cat(ℇ) of categories ℇ. In particular, the coefficient object is a category B in ℇ and the torsors that H¹ classifies are particular functors in ℇ. We only impose conditions on ℇ that are satisfied also by Cat(ℇ) and argue that H¹ for Cat(ℇ) is a kind of H² for ℇ, and so on recursively. For us, it is too much to ask ℇ to be a topos (or even internally complete) since, even if ℇ is, Cat(ℇ) is not. With this motivation, we are led to examine morphisms in ℇ 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 Β in ℇ 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.