The exponential map for vector fields on a manifold does not exist, at least not in any normal sense, i.e., as a mapping from vector fields to diffeomorphisms. The difficulty, of course, is that vector fields are not generally complete. Moreover, this lack of completeness is about as debilitating as one could hope, in that the infimum over the set of initial states of the length of the time interval on which an integral curve is defined is zero.

An appropriate substitute is devised for the exponential map, including in the formulation vector fields that depend on time and parameter. Two novel mathematical tools play a crucial role in the development. First, suitable topologies for sets of vector fields allow an elegant and uniform treatment of vector fields across varying regularity classes. Second, sheaves and groupoids for vector fields and diffeomorphisms allow for systematic localisation of the components of what will become the exponential map.

It will be illustrated that a complicated exponential map such as is described should play an essential role in problems of local controllability.

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