Quadratic functors on pointed categories |
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Authors: | Manfred Hartl Christine Vespa |
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Institution: | aUniversité de Lille Nord de France, F-59000 Lille, France;bUVHC, LAMAV and FR CNRS 2956, F-59313 Valenciennes, France;cUniversité de Strasbourg, Institut de Recherche Mathématique Avancée, UMR, 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France |
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Abstract: | We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups Ab, and whose source category is an arbitrary category C with null object such that all objects are colimits of copies of a generating object E which is small and regular projective; this includes all categories of models V of a pointed theory T. More specifically, we are interested in such quadratic functors F from C to Ab which preserve filtered colimits and suitable coequalizers.A functorial equivalence is established between such functors F:C→Ab and certain minimal algebraic data which we call quadratic C-modules: these involve the values on E of the cross-effects of F and certain structure maps generalizing the second Hopf invariant and the Whitehead product.Applying this general result to the case where E is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for C being the category of groups or of modules over some ring; here quadratic C-modules are equivalent with abelian square groups or quadratic R-modules, respectively. |
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Keywords: | MSC: 18D 18A25 55U |
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