Differential Operator, a Categorical approach

The aim of this essay is give an interesting view for differential operator $D$$D$.

Consider the Category ${\mathrm{DiffMan}}_{\ast }$$\mathrm{DiffMan_*}$. i.e., the object is differential manifold with base point, the morphism is smooth function preserve the base point.

Then $D$$D$ can be viewed as a functor,

$$\begin{array}{}\text{(1)}& D:{\mathrm{DiffMan}}_{\ast }⟶{\mathrm{Vect}}_{\mathbb{F}}\end{array}$$

Where $\mathbb{F}$$\mathbb F$ is $\mathbb{R}\text{ or }\mathbb{C}$$\mathbb R\text{ or }\mathbb C$.

$$\begin{array}{}\text{(2)}& D(M_p)=T_p(M),D(f)=Df(p)\end{array}$$

Then the chain rule

$$\begin{array}{}\text{(3)}& D(f\circ g)=Df(q)\circ Dg(p)\end{array}$$

Is just the property of functor.

I am wondering is that $D$$D$ is the left adjoint of the inclusion functor $I$$I$ from Category of ${\mathbb{R}}^n$$\mathbb R^n$ (As vector space) to ${\mathrm{DiffMan}}_{\ast }$$\mathrm{DiffMan}_*$?

i.e.

$$\begin{array}{}\text{(4)}& {\mathrm{Hom}}_{{\mathbb{R}}^n}(D(-),-)\cong {\mathrm{Hom}}_{{\mathrm{DiffMan}}_{\ast }}(-,I(-))\end{array}$$

But this is differential at the base point.

What if I want to discuss

$$\begin{array}{}\text{(5)}& D:C^{p+1}(M)\to C^p(N)\end{array}$$

or

$$\begin{array}{}\text{(6)}& d:{\mathrm{\Omega }}^p(U)\to {\mathrm{\Omega }}^{p+1}(U)\end{array}$$

?

Actually, there are Morphism of sheaf!

It is no hard to see that if you consider the funtor then you get an sheaf over $M$$M$

$$\begin{array}{}\text{(7)}& {{\mathrm{Hom}}^{\mathrm{p}}}_{\mathrm{Op}(M)}(-,\mathbb{R}):\mathrm{Op}(M)\to {\mathrm{Vect}}_{\mathbb{R}}\end{array}$$

For convenient, denote this functor as ${\mathcal{O}}_M^p$$\mathcal O^p_M$. $p\ge 0$$p\ge 0$ , and it could be $\mathrm{\infty }$$\infty$.

The reason we only consider the vector space structure $C^p(U)$$C^p(U)$ is $D$$D$ is not a ring homomorphism.

$$\begin{array}{}\text{(8)}& U⟼C^p(U),i_V^U⟼{\text{res}}_U^V\end{array}$$

Denote the Category of Sheaves over $M$$M$ as $\mathrm{Sh}(M)$$\mathrm{Sh}(M)$

Then $D$$D$ is the morphism betweem sheaves. i.e. Natural Transformation.

$$\begin{array}{}\text{(9)}& \begin{array}{c}\begin{array}{ccc}{\mathcal{O}}_M^{p+1}(V)& \stackrel{D_V}{\to }& {\mathcal{O}}_M^p(V)\\ {\mathrm{res}}_{\mathrm{U}}^{\mathrm{V}}\downarrow & & {\mathrm{res}}_{\mathrm{U}}^{\mathrm{V}}\downarrow & \\ {\mathcal{O}}_M^{p+1}(U)& \stackrel{D_U}{\to }& {\mathcal{O}}_M^p(V)\end{array}\end{array}\end{array}$$

Similarly, we could consider the sheaf of differential form.

$$\begin{array}{}\text{(10)}& {{\mathrm{Hom}}^{\mathrm{p}}}_{\mathrm{Op}(M)}(-,{\mathrm{Alt}}^{\mathrm{p}}({\mathbb{R}}^n)):\mathrm{Op}(M)\to {\mathrm{Vect}}_{\mathbb{R}}\end{array}$$

For convinience, denote this funcotr as ${\mathrm{\Omega }}_M^p$$\Omega^p_M$.

The element of ${\mathrm{\Omega }}_M^P(U)$$\Omega^P_M(U)$ is smooth function $\omega :U\to {\mathrm{Alt}}^{\mathrm{p}}({\mathbb{R}}^n)$$\omega:U\to \mathrm{Alt^p}(\mathbb R^n)$.

For example,

$$\begin{array}{}\text{(11)}& \omega =(3xy)\mathrm{d}x\wedge \mathrm{d}y-yz\mathrm{d}y\wedge \mathrm{d}z\end{array}$$

Then

$$\begin{array}{}\text{(12)}& \begin{array}{c}\begin{array}{ccc}{\mathrm{\Omega }}_M^p(V)& \stackrel{d_V}{\to }& {\mathrm{\Omega }}_M^{p+1}(V)\\ {\mathrm{res}}_{\mathrm{U}}^{\mathrm{V}}\downarrow & & {\mathrm{res}}_{\mathrm{U}}^{\mathrm{V}}\downarrow & \\ {\mathrm{\Omega }}_M^p(U)& \stackrel{d_U}{\to }& {\mathrm{\Omega }}^{p+1}M(U)\end{array}\end{array}\end{array}$$