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We continue the program of structural differential geometry that begins withthe notion of a tangent category, an axiomatization of structural aspects ofthe tangent functor on the category of smooth manifolds. In classical geometry,having an affine structure on a manifold is equivalent to having a flattorsion-free connection on its tangent bundle. This equivalence allows us todefine a category of affine objects associated to a tangent category and weshow that the resulting category is also a tangent category, as are severalrelated categories. As a consequence of some of these ideas we also give twonew characterizations of flat torsion-free connections.  We also consider 2-categorical structure associated to the category oftangent categories and demonstrate that assignment of the tangent category ofaffine objects to a tangent category induces a 2-comonad.  Finally, following work of Jubin, we consider monads and comonads on thecategory of affine objects associated to a tangent category. We show that thereis a rich theory of monads and comonads in this setting as well as variousdistributive laws and mixed distributive laws relating these monads andcomonads. Even in the category of smooth manifolds, several of these resultsare new or fill in gaps in the existing literature.

Affine geometric spaces in tangent categories