Norms on Categories and Analogs of the Schr\"oder-Bernstein Theorem

TL;DR

This paper introduces a generalized notion of norms on categories, unifying various mathematical areas and enabling new insights into metrization, with particular focus on the Schr"oder-Bernstein property and applications in topology and metric spaces.

Contribution

It formalizes a category-based norm concept that unifies diverse mathematical theories and incorporates the Schr"oder-Bernstein property as an axiom, facilitating new approaches to metrization.

Findings

Unified framework for norms across multiple mathematical disciplines.

Incorporation of Schr"oder-Bernstein property as an axiom.

Application to metrization and analysis of categories.

Abstract

We generalize the concept of a norm on a vector space to one of a norm on a category. This provides a unified perspective on many specific matters in many different areas of mathematics like set theory, functional analysis, measure theory, topology, and metric space theory. We will especially address the two last areas in which the monotone-light factorization and, respectively, the Gromov-Hausdorff distance will naturally appear. In our formalization a Schr\"oder-Bernstein property becomes an axiom of a norm which constitutes interesting properties of the categories in question. The proposed concept provides a convenient framework for metrizations.