§ Chain rule functorially

• The first category is $Euc^*$, whose objects are pointed subsets of Euclidean space. So, the objects are of the form $(U \subseteq \mathbb R^n, a \in U)$. Morphisms are based smooth functions between these opens: smooth functions $f: U \rightarrow V$such that $f(a) = b$.
• The second category is $M$whose objects are natural numbers and morphisms $n \to m$ are matrices of dimension $n \times m$.
• Define a functor $d: Euc \to M$ which sends subsets to the dimension of the space they live in: $U \subseteq \mathbb R^n, a) \to n$, and sends smooth functions $f: (U, a) \rightarrow (V, b)$ to their Jacobian evaluated at the basepoint, $J_f|_a$.
• Functoriality asserts that the jacobian of the composition of the two functions is a matrix multiplication of the jacobians.

Let's think through the functoriality. It says that if we have two arrows which can be composed, $(f, a)$ and $(g, f(a))$ -- note that it has to be $(g, f(a))$ because only compatible basepoint functions can be composed. I feel like we shouldn't use natural numbers for $M$, but we should rather use real vector spaces. And as for $Euc$, we should replace this with $ManOpen$ where we use based "charted" opens of a differentiable manifold. This makes the diffgeo clear!