# Differentiation Operator as a Function Between Spaces of Function Spaces

The differentiation operator $$D$$ takes a function $$f:A\to B$$ between differentiable manifolds $$A$$ and $$B$$ and assigns to it a function $$Df : A_0 \to \mathcal{L}(A,B)$$ which in turn assigns, to each point $$a_0\in A$$ at which $$f$$ is differentiable (we'll call the set of these points $$A_0$$), the total derivative of $$f$$ at $$a_0$$.

Now, I tried to "formally" define this operator with something along the lines of

$$D : \operatorname{Hom}_{\mathbf{Set}}(A, B) \to \operatorname{Hom}_{\mathbf{Set}} (A_0, \operatorname{Hom}_{\mathbb{R}\mathbf{-mod}}(A, B)),$$

but, needless to say, there seems to be much wrong with my attempt (it needs $$A$$, $$B$$, and, most tragically, somehow $$A_0$$, to be known in advance).

How could I go about "formally" writing down the definition I had written above in words in a compact (i.e. without any words), rigorous, category-theoretic (or so) way?

I'd want to do this without having to fix anything in advance, i.e. ideally we wouldn't be defining a scheme to define these operators, but rather just have "one operator to rule them all". Maybe someone with more experience in category theory or functional analysis can help me out, or point me in the right direction. Thank you!

• The first thing to notice is that $A$ and $B$ are not actually vector spaces. The differential of $f$ at a point $a$ is a linear application from the tangent space $T_a(A)$ to $T_{f(a)}(B)$. – Captain Lama Apr 25 at 15:24
• So do you mean $\operatorname{Hom}_{\mathbb{R}\mathbf{-mod}}(A,B)$ should be substituted with $\operatorname{Hom}_{\mathbb{R}\mathbf{-mod}}(T_a(A), T_{f(a)}(B))$? But (assuming that I'm not misunderstanding, which is a big assumption), that would still have the problem of explicitly needing $A$, $B$, and even $f$ be defined. – BlondCafé Apr 25 at 17:16
• Hmm... I think this might be the answer to my question (in which case my question would've been quite poorly formulated). – BlondCafé Apr 25 at 19:08
• Or, I suppose, maybe not; after all, at least in their presentation, $D$, whatever it might be, is limited to smooth (i.e. infinitely-differentiable) functions and manifolds. Any help or guidance (e.g. on differentiation functor vs differentiation operator vs differential operators) would be much appreciated! – BlondCafé Apr 25 at 21:35