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The comprehensive factorization and torsors

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Ross Street and Dominic Verity

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

Keywords:
torsor, internal category, exponentiable morphism, discrete
fibration; final functor, comprehensive factorization, locally isomorphic

2000 MSC:
18D35, 18A20, 14F19

*Theory and Applications of Categories,*
Vol. 23, 2010,
No. 3, pp 42-75.

http://www.tac.mta.ca/tac/volumes/23/3/23-03.dvi

http://www.tac.mta.ca/tac/volumes/23/3/23-03.ps

http://www.tac.mta.ca/tac/volumes/23/3/23-03.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/3/23-03.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/3/23-03.ps

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