### 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.

Language | English |
---|---|

Pages | 42-76 |

Number of pages | 35 |

Journal | Theory and Applications of Categories |

Volume | 23 |

Publication status | Published - 2010 |

### Fingerprint

### Cite this

}

*Theory and Applications of Categories*, vol. 23, pp. 42-76.

**The comprehensive factorization and torsors.** / Street, Ross; Verity, Dominic.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - The comprehensive factorization and torsors

AU - Street, Ross

AU - Verity, Dominic

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=76749131216&partnerID=8YFLogxK

M3 - Article

VL - 23

SP - 42

EP - 76

JO - Theory and Applications of Categories

T2 - Theory and Applications of Categories

JF - Theory and Applications of Categories

SN - 1201-561X

ER -