Quadratic functors on pointed categories

TL;DR

This paper characterizes quadratic polynomial functors from pointed categories to abelian groups, establishing an equivalence with algebraic data called quadratic modules, extending known results to broader categories.

Contribution

It introduces a functorial equivalence between quadratic functors and quadratic modules in general pointed categories, broadening the scope of previous specific cases.

Findings

Quadratic functors correspond to quadratic modules involving cross-effects.

The equivalence simplifies when the source object is a cogroup.

Extends results from groups and modules to more general 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 $F:\C\to Ab$ 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.