  I extend the framework of rigid analytic geometry to the setting of algebraic
geometry relative to monoids, and study the associated notions of separated,
proper, and overconvergent morphisms.
  The category of affine manifolds embeds as a subcategory defined by simple
algebraic (normal) and topological (overconvergent) criteria. The affine
manifold of a rigid space can be recovered either as a set of `Novikov field'
points or as a universal Hausdorff quotient. After base change to any
topological field, one obtains a `toric' analytic space that fibres over the
affine manifold.
