**Original manuscript:** 2022/07/20

The exponential map that characterises the flows of vector fields is the key in understanding the basic structural attributes of control systems in geometric control theory. However, this map does not exist due to the lack of completeness of flows for general vector fields.

An appropriate substitute is devised for the exponential map, not by trying to force flows to be globally defined by any compactness assumptions on the manifold, but by a categorical development of spaces of vector fields and flows, thus allowing for systematic localisation of such spaces. That is to say, we give a presheaf construction of the exponential map for vector fields with measurable time-dependence and continuous parameter-dependence in the category of general topological spaces. Moreover, all regularities in state are considered, including the cases of continuous, finitely differentiable, smooth and holomophic. Using geometric descriptions of suitable topologies for vector fields and for local diffeomorphisms, the homeomorphism of the exponential map is derived by a uniform treatment for these regularities. Finally, a new sort of continuous dependence is proved, that of the fixed time local flow on the parameter which plays an important role in the establishment of the homeomorphism of the exponential map.

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