Abstract

The Galois theory of Chase and Sweedler [11], for commutative rings, is generalized to encompass commutative monoids in an arbitrary symmetric, closed, monoidal category with finite limits and colimits. The primary tool is the Morita theory of Pareigis [35, 36, 37], which also supplies a suitable definition for the concept of a “finite” object in a monoidal category. The Galois theory is then extended by an examination of “normal” sub-Hopf-monoids, and examples in various algebraic and topological categories are considered. In particular, symmetric, closed, monoidal structures on various categories of topological vector spaces are studied with respect to the existence of “finite” objects.