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