22070 articles – 15901 Notices  [english version]
 HAL : hal-00334292, version 3
 arXiv : 0810.4502
 Versions disponibles : v1 (24-10-2008) v2 (11-11-2008) v3 (21-10-2009)
 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 pointed algebraic varieties. More specifically, we are interested in such quadratic functors $F$ from $\C$ to $Ab$ which preserve filtered colimits and suitable coequalizers; one may take reflexive ones if $\C$ is Mal'cev and Barr exact. A functorial equivalence is established between such functors $F:\C\to Ab$ 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.