Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on


* tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential map sends a vector to a point on a curve naturally defined in terms of the vector;
* unital Banach algebras?


(NB. I am not conversant with category theory beyond the words "morphism" and "functor". But a categorically flavored answer that takes my limited knowledge base into account would be preferable. An internet search led me to the notion of a "Cartesian closed category", which doesn't seem to be the sort of thing I have in mind.)



What is an exponential?
I think the exponential function for unital Banach algebras is a special case of the exponential functions for Lie groups modeled on topological vector spaces: the set of invertible elements in an unital Banach algebra naturally is a Banach Lie group, and the exponential function of this Lie group is the "classical" exponential function for the Banach algebra.


