The paper explores the assignment of norms to Λ-modules over a finite-dimensional algebra Λ, leading to the establishment of normed Λ-modules. The primary contribution is the construction of two new categories, Norp and its full subcategory A p, which provide a framework for understanding the categorification of integration, series expansions, and derivatives.
Key highlights:
The authors introduce the concepts of normed k-algebras and normed Λ-modules, along with their completions, to lay the foundation for the categorical approach.
They define the categories Norp and A p, where objects are triples consisting of a normed Λ-module, an element, and a specific Λ-homomorphism.
The authors prove that the triple (Sτ(IΛ), 1, γξ) is an A p-initial object in Norp, which enables the unique representation of normed modules.
The paper demonstrates how integrations, series expansions (including Taylor and Fourier series), and differentiations can be realized as morphisms in the category A p.
The authors establish the Stone-Weierstrass approximation theorem in the context of the category A p.
The results provide a unified categorical framework for understanding fundamental mathematical analysis concepts, potentially benefiting various scientific domains that rely on these techniques.
To Another Language
from source content
arxiv.org
Key Insights Distilled From
by Yu-Zhe Liu, ... at arxiv.org 10-03-2024
https://arxiv.org/pdf/2405.02777.pdfDeeper Inquiries