Affine geometric spaces in tangent categories

Abstract

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.