  We introduce a sheaf theoretic viewpoint on functional analysis designed for
infinite dimensional Lie group actions. We develop functional calculus for
Banach valued functors and, in particular, prove the existence of an
exponential map for a certain class of operators that generalise first order
partial differential operators. This algebraic framework can be used in
dynamical systems and KAM theory to provide normal forms and versal
deformations. it is used in our papers on the subject such as the Herman
conjecture paper, the versal deformation theorem for vector fields etc.
