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

Abstract

We explore the assignment of norms to Λ-modules over a finite-dimensional algebra Λ, resulting in the establishment of normed Λ-modules. Our primary contribution lies in constructing two new categories and, where each object in is a normed Λ-module N limited by a special element υN ∈ N and a special Λ-homomorphism, the morphism in is a Λ-homomorphism θ: N → M such that θ(υN) = υM and, and is a full subcategory of 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.