Quadratic functors on pointed categories
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 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 to Ab which preserve filtered colimits and suitable coequalizers; one may take reflexive ones if is Mal'cev and Barr exact. A functorial equivalence is established between such functors F: Ab and certain minimal algebraic data which we call quadratic -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 being the category of groups or of modules over some ring; here quadratic -modules are equivalent with abelian square groups or quadratic R-modules, respectively.