  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.
