Is there a natural category in which the morphisms are derivative operators?

Question

I'm studying general relativity. As I currently understand the theory, there's a part where we have a (differentiable) manifold $M$, and define a vector field on $M$ to be a function $v : (M \to \Re) \to (M \to \Re)$ satisfying:

1. (linearity) $v(\alpha f + \beta g) = \alpha v(f) + \beta v(g)$
2. (Leibniz) $v(f \cdot g) = f \cdot v(g) + g \cdot v(f)$

we might also need to further restrict our attention to $v$ that are continuous. We can then show that the set of functions $v$ that satisfy these properties form a vector space, at which point we can generalize from vector fields to tensor fields. (Then, given a metric on $M$ we can derive a natural notion of differentiation on tensor fields, at which point we're ready to state some properties that the spacetime metric and the stress-energy tensor obey.)

My question is, is there a (natural) category in which vector fields are just endomorphisms on $(M \to \Re)$? For example, condition (1) above arises automatically if we require that $v$ be an endomorphism of $(M \to \Re)$ in $\Re$-Vect; is there a well-known category such that conditions (1) and (2) arise automatically if we require $v$ to be an endomorphism of $(M \to \Re)$?

Obviously I could simply define a category where the objects are vector spaces and the morphisms are linear maps that happen to satisfy the Leibniz rule [EDIT: This is wrong, as pointed out by Eric below -- given two $v$ that satisfy the property above, their composition does not in general satisfy the Leibniz property]; my question is, is this a well-known category (or, is there a simple variation on my question that allows me to see vector fields in a categorical light)?

As an example of the genre of question that I'm asking, recall that a vector space over a field $K$ with vectors $V$ can be viewed either as a function $* : K \to V \to V$ satisfying a handful of axioms, or simply as a ring homomorphism between $K$ and the ring of group endomorphisms on $V$. In other words, we can either specify a vector space as a function obeying a bunch of axioms, or we can choose a morphism between the right objects in the right category (in this case, any $\phi : K \to_\text{Ring} (V \to_\text{Group} V)$) at which point the axioms come free. In the case of vector spaces, I could have defined a category in which all morphisms are vector spaces, but I probably wouldn't have noticed that I was trying to ask for a ring homomorphism between the scalar field and the ring of endomorphisms of the vector group. In my question here about a category where morphisms correspond to derivative operators, I'm hoping for an answer analogous to "you're looking for a ring homomorphism between $K$ and $V \to_\text{Group} V$".

Answer 1

Instead of talking about manifolds let's talk for a bit about vector fields on affine schemes. For the sake of minimizing technical detail my definition will be that the category $\text{Aff}$ of affine schemes is just the opposite of the category $\text{CRing}$ of commutative rings, and if $R$ is a commutative ring then $\text{Spec } R$ will just be my name for $R$ but regarded as an object in the opposite category.

Definition: A vector field on $\text{Spec } R$ is a derivation $D : R \to R$; that is, a linear function satisfying $D(ab) = a D(b) + D(a) b$ for all $a, b \in R$.

The intuition here, as in the manifold case, is that derivations are "infinitesimal automorphisms" of $R$, or equivalently of $\text{Spec } R$. The lovely benefit of switching to working with affine schemes is that it is possible to make this intuition completely precise as follows.

Exercise: A derivation $D$ on a commutative ring $R$ is the same thing as an $\epsilon$-linear homomorphism $R[\epsilon]/\epsilon^2 \to R[\epsilon]/\epsilon^2$ which reduces to the identity $\bmod \epsilon$.

In particular, the Leibniz rule is equivalent to the assertion that the map $I + \epsilon D$ preserves multiplication, and composition of homomorphisms as above corresponds to addition of derivations. See this blog post for more on what can be done from this perspective, including a clean conceptual proof that the commutator of two derivations is a derivation.

Geometrically the above definition corresponds to asking for a map from $\text{Spec } R \times \text{Spec } \mathbb{Z}[\epsilon]/\epsilon^2$ to $\text{Spec } R$; you can think of $\text{Spec } \mathbb{Z}[\epsilon]/\epsilon^2$ as the "walking tangent vector" and of this definition as asking for a tangent vector to the identity in the "space" (it would be an affine scheme if the category of affine schemes were cartesian closed, which it's not; however it is a presheaf on the category of affine schemes, just not a representable one) of endomorphisms of $\text{Spec } R$.

More generally you can contemplate homomorphisms $R \to S[\epsilon]/\epsilon^2$ for two rings $R$ to $S$, or equivalently $\epsilon$-linear homomorphisms $R[\epsilon]/\epsilon^2 \to S[\epsilon]/\epsilon^2$; this works out to a map of the form $f + \epsilon g$ where $f : R \to S$ is a homomorphism and $g : R \to S$ is a linear map satisfying

$$g(ab) = f(a) g(b) + g(a) f(b).$$

This is a useful generalization of the notion of a derivation; for example it can be used to define Zariski tangent spaces. It can be thought of as describing a tangent vector to $f$ in the "space" of maps from $R$ to $S$, or equivalently from $\text{Spec } S$ to $\text{Spec } R$. The corresponding object in differential geometry is a pair consisting of a map $f : M \to N$ of smooth manifolds and a section of the pullback $f^{\ast}(TN)$ of the tangent bundle of $N$ to $M$.

You can try to make this sort of thing work out for smooth manifolds by passing to a somewhat more complicated category that includes smooth manifolds as well as various "infinitesimal" spaces; see smooth algebra for some details.

On the more general topic of categorical approaches to tensor fields, you might enjoy skimming Kolar, Michor, and Slovak's Natural Operations in Differential Geometry.