Quadratic functors on pointed categories
Résumé
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 T with null object such that all objects are finite coproducts of a generating object E. A functorial equivalence is established between quadratic functors F from T to Ab and certain minimal algebraic data which we call quadratic T-modules: these involve the values on E of the cross-effects of F and certin structure maps generalizing the second Hopf invariant and the Withehead product. Applying this general result to the case where E is a cogroup these data take a particulary simple form. This application extends results of Baues and Pirashvili obtained for T being the category of free groups or modules of finite rank; here quadratic T-modules are equivalent with square groups or quadratic R-modules, respectively.
Origine : Fichiers produits par l'(les) auteur(s)