  We show that the functor of $p$-typical co-Witt vectors on commutative
algebras over a perfect field $k$ of characteristic $p$ is defined on, and in
fact only depends on, a weaker structure than that of a $k$-algebra. We call
this structure a $p$-polar $k$-algebra. By extension, the functors of points
for any $p$-adic affine commutative group scheme and for any formal group are
defined on, and only depend on, $p$-polar structures. In terms of abelian Hopf
algebras, we show that a cofree cocommutative Hopf algebra can be defined on
any $p$-polar $k$-algebra $P$, and it agrees with the cofree commutative Hopf
algebra on a commutative $k$-algebra $A$ if $P$ is the $p$-polar algebra
underlying $A$; a dual result holds for free commutative Hopf algebras on
finite $k$-coalgebras.
