Affine functors and duality

A functor of sets $\mathbb{X}$ over the category of $K$-commutative algebras is said to be an affine functor if its functor of functions, $\mathbb A_{\mathbb X}$,isreflexiveand$\mathbb X=\Spec \mathbb A_{\mathbb X}$. We prove that affine functors are equal to a direct limit of affine schemes and that affine schemes, formal schemes, the completion of affine schemes along a closed subscheme, etc., are affine functors. Endowing an affine functor $\mathbb{X}$ with a functor of monoids structure is equivalent to endowing ${\mathbb{A}}_{\mathbb{X}}$ with a functor of bialgebras structure. If $\mathbb{G}$ is an affine functor of monoids, then $\mathbb A_{\mathbb G}^$istheenvelopingfunctorofalgebrasof$\mathbb G$ and the category of $\mathbb{G}$-modules is equivalent to the category of $\mathbb A_{\mathbb G}^$-modules. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes, the equivalence between formal groups and Lie algebras in characteristic zero, etc.