Quadratic functors on pointed categories

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.

💡 Research Summary

The paper develops a general theory of quadratic (degree‑2 polynomial) functors from an arbitrary pointed category C to the category of abelian groups, under very mild structural assumptions on C. The authors require that C possess a null object and a distinguished generating object E which is small, regular projective, and such that every object of C can be expressed as a colimit of copies of E. This setting includes all pointed algebraic varieties, the category of groups, and categories of modules over a ring, among many others.

A quadratic functor F:C→Ab is defined by the vanishing of its third and higher cross‑effects, i.e. crₙF=0 for n≥3, while cr₂F may be non‑trivial. The paper focuses on those quadratic functors that preserve filtered colimits and a class of coequalizers (reflexive coequalizers when C is Mal’cev and Barr exact). Under these preservation hypotheses, the authors show that the entire behaviour of F is encoded by a very small amount of data attached to the generator E.

Specifically, they introduce the notion of a “quadratic C‑module”. Such a module consists of:

1. The value M₁ = F(E) (the linear part).
2. The value M₂ = cr₂F(E,E) (the quadratic cross‑effect).
3. A structure map ψ : M₂ → M₁, which generalises the second Hopf invariant and records how two copies of E interact when they are summed.
4. A bilinear map ω : Λ²M₂ → M₂ (or an equivalent formulation), which plays the role of a Whitehead product, measuring the non‑commutativity of the quadratic interaction.

These maps are required to satisfy a collection of axioms expressing associativity, symmetry, and compatibility with the abelian group structures. When the generator E carries a cogroup structure, ψ and ω become especially simple: ψ is induced by the cogroup comultiplication and ω by the cogroup’s interchange law. In this special case the quadratic C‑module coincides with the classical objects known as abelian square groups (for C = groups) or quadratic R‑modules (for C = R‑modules).

The central theorem establishes an equivalence of categories:

 Quadratic functors F:C→Ab preserving the chosen colimits ↔ Quadratic C‑modules (M₁,M₂,ψ,ω).

The proof proceeds in two directions. From a functor F, one extracts M₁ and M₂ via the first and second cross‑effects, and the preservation properties guarantee that the values on arbitrary objects are determined by the values on the generating copies of E. The maps ψ and ω are obtained by evaluating F on the canonical codiagonal and on the canonical “commutator” morphisms coming from the cogroup structure of E (or from the universal properties of coproducts). Conversely, given a quadratic C‑module, the authors construct a functor F by first defining it on free objects (coproducts of copies of E) using the prescribed data, then extending uniquely to all objects by filtered colimit preservation. The axioms on ψ and ω ensure that the construction respects the required coequalizers and yields a well‑defined functor.

After establishing the equivalence, the paper works out several concrete instances. When C is the category of groups and E = ℤ, a quadratic C‑module is precisely an abelian square group, recovering Baues’s description of quadratic group functors. When C is the category of left R‑modules and E = R, the quadratic C‑module becomes a quadratic R‑module, matching Pirashvili’s earlier results. The authors also discuss how the framework applies to other pointed algebraic varieties such as Lie algebras, associative algebras, or more exotic structures, illustrating the broad reach of the theory.

In the concluding section the authors point out that the same methodology should extend to higher‑degree polynomial functors. One expects a hierarchy of “higher‑order C‑modules” encoding the data of crₙF for n>2, together with higher analogues of ψ and ω. Such a development would provide a unified categorical language for studying secondary and tertiary operations in homotopy theory, cohomology operations, and deformation theory. The paper therefore not only generalises known results for groups and modules but also lays a solid foundation for future investigations of polynomial functors in a wide variety of algebraic contexts.