The comprehensive factorization and torsors

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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.

LanguageEnglish
Pages42-76
Number of pages35
JournalTheory and Applications of Categories
Volume23
Publication statusPublished - 2010

Fingerprint

Torsor
Factorization
Functor
Classify
Internal
Topos
Fibration
Cocycle
Coefficient
Morphisms
Theorem
Cohomology
Isomorphic

Cite this

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title = "The comprehensive factorization and torsors",
abstract = "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.",
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The comprehensive factorization and torsors. / Street, Ross; Verity, Dominic.

In: Theory and Applications of Categories, Vol. 23, 2010, p. 42-76.

Research output: Contribution to journalArticleResearchpeer-review

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