# What is an exponential?

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.)

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Exponentiation can mean 1. the function $x \rightarrow \exp(x)$ or 2. the function $(a,b) \rightarrow a^b$. Cartesian closed categories are about the latter. – Dan Piponi Jan 23 '10 at 15:25
That's why it didn't seem like it to me. – Steve Huntsman Jan 23 '10 at 15:54

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.

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Nice, but if there is a group structure on generic (nice, finite-dimensional) Riemannian manifolds I don't know about it. – Steve Huntsman Jan 26 '10 at 21:50
@Hanno Becker @Steve Huntsman just a note: the group of invertible elements in a Banach algebra has natural structure of manifold modeled on the underlying Banach space, and I know Serge Lang's Foundamentals of Differential Geometry as the reference for Banach Manifolds. There the exponential map is associated to any spray on a Banach manifold. – Giuseppe Apr 16 '11 at 20:41