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 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 → 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-

Foncteurs Faiblement Polynomiaux

“…Depuis, de nombreuses recherches autour des foncteurs polynomiaux se sont développées. Plusieurs travaux concernent l'étude des foncteurs polynomiaux d'un point de vue intrinsèque [35,25,47,23]. D'autres travaux sont motivés par l'utilisation de ces foncteurs en topologie algébrique, qui a connu un renouveau au début des années 1990 grâce à l'article [24] de Henn, Lannes et Schwartz reliant la catégorie des modules instables sur l'algèbre de Steenrod à ces foncteurs.…”

“…Même si les foncteurs polynomiaux entre différentes catégories de modules sont restés le cas le plus étudié, plusieurs de ces travaux utilisent des foncteurs polynomiaux dans d'autres contextes [38,23,5]. Ces généralisations reposent toujours sur la notion d'effets croisés inaugurée par Eilenberg et Mac Lane.…”

Central reflections and nilpotency in exact Mal’tsev categories

“…Special instances of the following theorem have been considered in literature: if E is the category of groups and n = 2, the result can be deduced from Baues-Pirashvili [1] and Hartl-Vespa [40]; if E is a semi-abelian category, under the identification of n-folded objects given in Remark 6.4 and with the kernel intersection definition of degree, the result has been announced by Manfred Hartl in several of his talks. Theorem 6.8.…”

“…There are several other places in literature where degree n functors, especially quadratic functors, are studied in a non-additive context, most notably Baues-Pirashvili [1], Johnson-McCarthy [50] and Hartl-Vespa [40]. In all these places, the definition of a degree n functor is based on a vanishing condition of cross-effects, closely following the original approach of Eilenberg-Mac Lane.…”

Sur l’homologie des groupes d’automorphismes des groupes libres à coefficients polynomiaux

“…Pour la partie droite du diagramme : pour F ∈ Pol n ( E C ), on a un isomorphisme naturel S n -équivariant Hom( [14] pour une démonstration détaillée dans le cas n = 2). Cet isomorphisme induit un…”