Affine Functors of Monoids and Duality

Let G = SpecA be an affine functor of monoids. We prove that A∗ is the enveloping functor of algebras of G and that the category of Gmodules is equivalent to the category of A∗-modules. Moreover, we prove that the category of affine functors of monoids is anti-equivalent to the category of functors of affine bialgebras. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes and the equivalence between formal groups and Lie algebras in characteristic zero. Finally, we also show how these results can be used to recover and generalize some aspects of the theory of the Reynolds operator.