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

TL;DR

This work develops a categorical framework for analysis over finite-dimensional $\mathds{k}$-algebras by introducing normed $\Lambda$-modules and the categories $\mathscr{Nor}^p$ and its Banach-subcategory $\mathscr{A}^p$. It identifies an $\mathscr{Nor}^p$-initial object $(\mathbf{S}_{\tau}(\mathbb{I}_{\Lambda}), \mathbf{1}, \gamma_{\xi})$ that yields a universal morphism to any normed module, enabling a unified representation of normed representations of algebras and a categorified Lebesgue integration via a morphism $\widehat{T}$. The paper then realizes series expansions as morphisms in $\mathscr{A}^p$: power-series and Fourier-series via $E_{\mathrm{pow}}$ and $E_{\mathrm{Fou}}$, with the Weierstrass theorem emerging from these constructions; differentiation is treated in $\mathscr{A}^p$ where a derivative morphism $D$ can be defined, while it cannot originate from the initial object in $\mathscr{A}^1$. These results illuminate deep connections between algebraic structure and classical analysis, providing a framework with potential implications for computation and mathematical physics.

Abstract

We explore the assignment of norms to $\mathitΛ$-modules over a finite-dimensional algebra $\mathitΛ$, resulting in the establishment of normed $\mathitΛ$-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 $\mathitΛ$-module $N$ limited by a special element $v_N\in N$ and a special $\mathitΛ$-homomorphism $δ_N: N^{\oplus 2^{\dim\mathitΛ}} \to N$, the morphism in $\mathscr{N}\!\!or^p$ is a $\mathitΛ$-homomorphism $θ: N\to M$ such that $θ(v_N) = v_M$ and $θδ_N = δ_Mθ^{\oplus 2^{\dim\mathitΛ}}$, 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 $\mathscr{A}^p$.

Figures (1)

• Figure 3.1: Juxtaposition map

Theorems & Definitions (105)

• Theorem 1.2: Theorem \ref{['thm:main1']} and Remark \ref{['rmk:main1']}
• Theorem 1.3: Theorem \ref{['thm:main2']}
• Corollary 1.4
• Corollary 1.5: Corollary \ref{['coro:S-W approx']}, Stone$-$Weierstrass Approximation Theorem
• Theorem 1.6
• Definition 2.1: cf. R1979
• Example 2.2
• Definition 2.3: cf. R1979
• Example 2.4
• Definition 2.5