The comprehensive factorization and torsors

Ross Street*, Dominic Verity

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


This is an expanded, revised and corrected version of the first author's 1981 preprint[1]. The discussion of one-dimensional cohomology H1 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 H1 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 H1 for Cat(E{open}) is a kind of H2 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 languageEnglish
Pages (from-to)42-76
Number of pages35
JournalTheory and Applications of Categories
Publication statusPublished - 2010


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