  The classical notion of retraction map used to approximate geodesics is
extended and rigorously defined to become a powerful tool to construct
geometric integrators and it is called discretization map. Using the geometry
of the tangent and cotangent bundles, we are able to tangently and cotangent
lift such a map so that these lifts inherit the same properties as the original
one and they continue to be discretization maps. In particular, the cotangent
lift of a discretization map is a natural symplectomorphism, what plays a key
role for constructing geometric integrators and symplectic methods. As a
result, a wide range of (higer-order) numerical methods are recovered and
canonically constructed by using different discretization maps, as well as some
operations with Lagrangian submanifolds.
