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First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due to Grothendieck:

Let $A$ be a commutative algebra, and $X$ an $A$-module. Then the differential operators on $X$ is the filtered algebra $D = D(A,X)$ given inductively by: $$D_{\leq 0} = \text{image of A in }\hom_k(X,X)$$ $$D_{\leq n} = \{ \phi \in \hom_k(X,X) \text{ s.t. } [\phi,a] \in D_{\leq n-1}\, \forall a\in A\}$$ $$D = \bigcup_{n=0}^\infty D_{\leq n}$$ (Edit: In the comments, Michael suggests that $D_{\leq 0} = \hom_A(X,X)$ is the more standard definition, and the rest is the same.)

Then the following facts are more or less standard:

• If $A$ acts freely on $X$, then $D_{\leq 1}$ acts on $A$ by derivations. (It's always true that $D_{\leq 1}$ acts on $D_{\leq 0}$ by derivations; the question is whether $D_{\leq 0} = A$ or a quotient.) If $X = A$ by multiplication, then $D_{\leq 1}$ splits as a direct sum $D_{\leq 1} = \text{Der}(A) \oplus A$.
• If $k=\mathbb R$, $M$ is a finite-dimensional smooth manifold, and $A = C^\infty(M) = X$, then $D$ is the usual algebra of differential operators generated by $A$ and $\text{Vect}(M) = \Gamma(TM \to M)$.
• If $A,X$ are actually sheaves, so is $D$.

Thus, at least in the situation where $A = X = C^{\infty}(M)$, the algebra $D$ is acting very much like the universal enveloping algebra of $U (\text{Vect}(M))$; in particular, the map $U(\text{Vect}(M)) \to D$ is filtered and is (almost) a surjection: it misses only the non-constant elements of $A$. So when $A = X = C^\infty(-)$ are sheaves on $M$, it's very tempting to think of $D$ as a sheafy version of $U(\text{Vect}(-))$. Note that $U(\text{Vect}(-))$ is not a sheaf: its degree $\leq 0$ part consists of constant functions, not locally constant, for example, and there are non-zero elements in $U_{\leq 2}$ that restrict to $0$ on an open cover. I think that it cannot be true that the sheafification of $U(\text{Vect}(-))$ is $D$, as the sheafification of $U_{\leq 0}$ is the locally-constant sheaf, not $C^\infty$.

So: is there a description of $D$ that makes it more obviously like a universal enveloping algebra? E.g. is there some adjunction or other categorical description? Is it really true that $D$ is a "sheafy" version of $U$ in a precise sense, or is this just a chimera?

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# Is there a "categorical" description of Grothendieck's algebra of differential operators?

First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due to Grothendieck:

Let $A$ be a commutative algebra, and $X$ an $A$-module. Then the differential operators on $X$ is the filtered algebra $D = D(A,X)$ given inductively by: $$D_{\leq 0} = \text{image of A in }\hom_k(X,X)$$ $$D_{\leq n} = \{ \phi \in \hom_k(X,X) \text{ s.t. } [\phi,a] \in D_{\leq n-1}\, \forall a\in A\}$$ $$D = \bigcup_{n=0}^\infty D_{\leq n}$$

Then the following facts are more or less standard:

• If $A$ acts freely on $X$, then $D_{\leq 1}$ acts on $A$ by derivations. (It's always true that $D_{\leq 1}$ acts on $D_{\leq 0}$ by derivations; the question is whether $D_{\leq 0} = A$ or a quotient.) If $X = A$ by multiplication, then $D_{\leq 1}$ splits as a direct sum $D_{\leq 1} = \text{Der}(A) \oplus A$.
• If $k=\mathbb R$, $M$ is a finite-dimensional smooth manifold, and $A = C^\infty(M) = X$, then $D$ is the usual algebra of differential operators generated by $A$ and $\text{Vect}(M) = \Gamma(TM \to M)$.
• If $A,X$ are actually sheaves, so is $D$.

Thus, at least in the situation where $A = X = C^{\infty}(M)$, the algebra $D$ is acting very much like the universal enveloping algebra of $U (\text{Vect}(M))$; in particular, the map $U(\text{Vect}(M)) \to D$ is filtered and is (almost) a surjection: it misses only the non-constant elements of $A$. So when $A = X = C^\infty(-)$ are sheaves on $M$, it's very tempting to think of $D$ as a sheafy version of $U(\text{Vect}(-))$. Note that $U(\text{Vect}(-))$ is not a sheaf: its degree $\leq 0$ part consists of constant functions, not locally constant, for example, and there are non-zero elements in $U_{\leq 2}$ that restrict to $0$ on an open cover. I think that it cannot be true that the sheafification of $U(\text{Vect}(-))$ is $D$, as the sheafification of $U_{\leq 0}$ is the locally-constant sheaf, not $C^\infty$.

So: is there a description of $D$ that makes it more obviously like a universal enveloping algebra? E.g. is there some adjunction or other categorical description? Is it really true that $D$ is a "sheafy" version of $U$ in a precise sense, or is this just a chimera?