We have natural functors:
$ Mfd\hookrightarrow Top ~~~~$ from the category of smooth manifolds to that of topological spaces,
$ LieGrp\hookrightarrow TopGrp ~~~~$ from the category of Lie groups to that of topological groups.
So I am wondering if these functors admit left adjoints? I guess that there are not such adjoints. However, I can not prove or disprove. I also have seen adjoint functor theorems, but I don't know how to apply them for these particular cases. Perhaps some necessary condition could be useful to disprove.
In the case of spaces, note that $L(*)$ must again be a point, by the adjunction condition. Now $L$ would preserve coproducts, but arbitrary coproducts, even of a point, do not exist in the category of smooth manifolds. (To be sure of this, just note that every smooth manifold has cardinality at most that of the continuum, and consider maps from a big coproduct of points to the two-point manifold.) A similar argument works for groups, considering instead $\mathbb Z$.