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THE COMPREHENSIVE FACTORIZATION AND TORSORS
, 2010
"... This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the tors ..."
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This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the torsors that H1 classifies are particular functors in E. We only impose conditions on E that are satisfied also by Cat(E) and argue that H1 for Cat(E) is a kind of H2 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 nal functor followed by a discrete bration. We de ne Btorsors for a category B in E and prove clutching and classification theorems. The former theorem clutches Ć’ech 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.
Logical Construction of Final Coalgebras
, 2003
"... We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object. ..."
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We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object.