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 constructingtwo new categories $\mathscr{N}\!\!or^p$ and $\mathscr{A}^p$, whereeach 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 $\mathscr{A}^p$.