### Abstract

This is an expanded, revised and corrected version of the first author's 1981 preprint[1]. The discussion of one-dimensional cohomology H^{1} in a fairly general category E{open} involves passing to the (2-)category Cat(E{open}) of categories in E{open}. In particular, the coefficient object is a category B in E{open} and the torsors that H^{1} classifies are particular functors in E{open}. We only impose conditions on E{open} that are satisfied also by Cat(E{open}) and argue that H^{1} for Cat(E{open}) is a kind of H^{2} for E{open}, and so on recursively. For us, it is too much to ask E{open} to be a topos (or even internally complete) since, even if E{open} is, Cat(E) is not. With this motivation, we are led to examine morphisms in E{open} 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{open} and prove clutching and classification theorems. The former theorem clutches Cech 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.

Original language | English |
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Pages (from-to) | 42-76 |

Number of pages | 35 |

Journal | Theory and Applications of Categories |

Volume | 23 |

Publication status | Published - 2010 |