# The comprehensive factorization and torsors

## 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

TAC Home