Quadratic functors on pointed categories

Manfred Hartl (1) , Christine Vespa (2)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015
2 IRMA - Institut de Recherche Mathématique Avancée

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.

Mots clés: polynomial functors; quadratic functors; algebraic theory

Domaines: Topologie algébrique [math.AT]; Catégories et ensembles [math.CT]; Théorie des groupes [math.GR]

Manfred Hartl, Christine Vespa. Quadratic functors on pointed categories. Advances in Mathematics, 2011, 226 (5), pp.3927-4010. ⟨10.1016/j.aim.2010.11.008⟩. ⟨hal-00334292v3⟩