Does the category of smooth manifolds over $\mathbb{R}$ have an initial object?
And if not, is there a natural way to transform this category so that it has an initial object.
I think that the zero-dimensional manifold which should be the same as the trivial vector space is a final object. But I don't see how to map this final object to a smooth manifold. So I would think that there is no initial object.
If $e$ is the initial object of a category $C$, for every object $X$ of $C$ there exists a unique morphism $i_X:e\rightarrow X$.
Suppose that $e$ is the initial object of $Mani$ the category of manifolds, if $X$ has more than two points, and $e$ is not empty, there are more than two distinct maps $e\rightarrow X$.Therefore there exists an initial object if it is assumed that the empty set is a manifold.