Multiplicity of a space over another space

TL;DR

This paper introduces the concept of multiplicity in various mathematical categories, defining a pseudo-distance based on it, and explores its properties across topological spaces, groups, modules, and knots.

Contribution

It formalizes the notion of multiplicity for morphisms and objects, and applies it to define a pseudo-distance in multiple mathematical categories.

Findings

Defined multiplicity of morphisms and objects.

Established a pseudo-distance on classes of objects.

Analyzed multiplicities in topological spaces, groups, modules, and knots.

Abstract

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. We define and study several multiplicities in the category of topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated $R$-modules and $R$-linear maps over a principal ideal domain $R$, and the neighbourhood category of oriented knots in the 3-sphere.