Can the exponential map (in Lie theory) be considered an example of a functor?

Question

I've heard that the exponential map from the Lie algebra of a Lie group to the original Lie group can be considered as a functor of some sort, but I'm very new to all this and would appreciate some clarification. If this is true, a brief explanation of what exactly this functor is would be much appreciated.

Answer 1

The exponential map is not a functor, but a natural transformation. This is a formalization that, if $\varphi:G\to H$ is a map of Lie groups and $D\varphi:{\mathfrak g}\to {\mathfrak h}$ is its derivative viewed as a map of Lie algebras, then we have an equality ${\rm exp}_H\circ D\varphi=\varphi\circ {\rm exp}_G$ of maps $\mathfrak g\to H$. The functors at issue here are the forgetful functor from Lie groups to smooth manifolds and the functor from Lie groups to smooth manifolds sending $G\mapsto \mathfrak g$ and $\varphi\mapsto D\varphi$.

Answer 2

There are non-isomorphic Lie groups that have isomorphic Lie algebras -- for example take $SL_2(\mathbb{R})$ and $PSL_2(\mathbb{R}) = SL_2(\mathbb{R})/\{\pm1\}$. So certainly you cannot expect a functor that always returns the group you started with.

You can however rectify this by say requiring the Lie group to be simply connected.