TITLE: Universal property of the tangent bundle
QUESTION [28 upvotes]: If $X$ is a scheme (over some base scheme, but which I will ignore) its tangent bundle $T(X)$ is defined as the relative spectrum of the symmetric algebra of its sheaf of differentials. Combining the universal properties of these three constructions, we get a universal property of $T(X)$, namely: Defining $U[\epsilon] := U \times \text{Spec}(\mathbb{Z}[\epsilon]/\epsilon^2)$, we have a canonical bijection
$Hom(U,T(X)) \cong Hom(U[\epsilon],X)$
Thus $T(-)$ is right adjoint to $(-)[\epsilon]$.
Question Is there a similar universal property of the tangent bundle of a manifold?
Here a manifold is assumed to be smooth and paracompact. I doubt that we can literally translate it, because $U[\epsilon]$ is $U$ as a topological space, but the structure sheaf now has the nilpotent $\epsilon$. My motivation comes from the observation that in every construction of the tangent bundle of a manifold I know, some nasty calculations with charts have to be made. I want to avoid this with the help of a universal property, proving that $T(\mathbb{R}^n)$ exists and then formally deducing the existence of $T(M)$ for every manifold. I know this can be done without using a universal property and that this would possibly not be the best construction or characterization of the tangent bundle, but it hopefully avoids irrelevant choices of charts. A similar question was asked here and here, but this seems to go in another direction.

REPLY [10 votes]: The algebro-geometric definition does translate, almost literally.  John Klein's comment to Martin's question provides one way of doing the translation.
Let $X$ be a manifold.  Define a contravariant functor $F$ on manifolds by taking $F(Y)$ to be the set of pairs $(f, \delta)$ where $f : Y \rightarrow X$ is a morphism of manifolds and $\delta : C^\infty(X) \rightarrow C^\infty(Y)$ is a derivation with respect to the module structure on $C^\infty(Y)$ induced from $f$.  Then $F$ is representable by $TX$.
Alternately, you may think of an element of $F(Y)$ as an extension of $f : Y \rightarrow X$ to a map of ringed spaces $Y[\epsilon] \rightarrow X$ where $Y[\epsilon]$ is the space $Y$, ringed by $C^\infty(Y)[\epsilon] / (\epsilon^2)$.