Normed modules and the categorification of integrations, series expansions, and differentiations

We explore the assignment of norms to $\mathit{\Lambda}$-modules over a finite-dimensional algebra $\mathit{\Lambda}$, resulting in the establishment of normed $\mathit{\Lambda}$-modules. Our primary contribution lies in constructing two new categories $\mathscr{N}\!\!or^p$ and $\mathscr{A}^p$, where each object in $\mathscr{N}\!\!or^p$ is a normed $\mathit{\Lambda}$-module $N$ limited by a special element $v_N\in N$ and a special $\mathit{\Lambda}$-homomorphism $\delta_N: N^{\oplus 2^{\dim\mathit{\Lambda}}} \to N$, the morphism in $\mathscr{N}\!\!or^p$ is a $\mathit{\Lambda}$-homomorphism $\theta: N\to M$ such that $\theta(v_N) = v_M$ and $\theta\delta_N = \delta_M\theta^{\oplus 2^{\dim\mathit{\Lambda}}}$, and $\mathscr{A}^p$ is a full subcategory of $\mathscr{N}\!\!or^p$ generated by all Banach modules. By examining the objects and morphisms in these categories. We establish a framework for understanding the categorification of integration, series expansions, and derivatives. Furthermore, we obtain the Stone--Weierstrass approximation theorem in the sense of $\mathscr{A}^p$.