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

We explore the assignment of norms to $\Lambda$-modules over a finite-dimensional algebra $\Lambda$, resulting in the establishment of normed $\Lambda$-modules. Our primary contribution lies in constructing two new categories $\mathcal{N}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}o{r}^p$ and ${\mathcal{A}}^p$, where each object in $\mathcal{N}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}o{r}^p$ is a normed $\Lambda$-module $N$ limited by a special element $v_N\in N$ and a special $\Lambda$-homomorphism ${\delta }_N:N^{\oplus 2^{\dim \Lambda }}\to N$, the morphism in $\mathcal{N}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}o{r}^p$ is a $\Lambda$-homomorphism $\theta :N\to M$ such that $\theta (v_N)=v_M$ and $\theta {\delta }_N={\delta }_M{\theta }^{\oplus 2^{\dim \Lambda }}$, and ${\mathcal{A}}^p$ is a full subcategory of $\mathcal{N}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}o{r}^p$ 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 ${\mathcal{A}}^p$.