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

TL;DR

This paper develops new categorical frameworks for normed modules over finite-dimensional algebras, enabling a categorification of classical analysis concepts like integration and differentiation, and proves an approximation theorem within this setting.

Contribution

It introduces categories of normed modules with special structures, facilitating the categorification of integration, series expansions, and derivatives, and establishes a Stone--Weierstrass type approximation theorem.

Findings

Constructed categories $ ext{Nor}^p$ and $ ext{A}^p$ for normed modules.

Established a framework linking categorical structures to classical analysis.

Proved a Stone--Weierstrass approximation theorem in the categorical context.

Abstract

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 $\mathscr{A}^p$.