The concept of null in general spaces and contexts

TL;DR

This paper develops a generalized, functorial framework for nullity in various mathematical contexts, unifying notions of null-sets across topology, measure theory, and other areas.

Contribution

It introduces a formal, categorical approach to nullity, extending existing notions via Kan extensions to broader mathematical structures.

Findings

Nullity can be formalized as a functor between categories.

A constructive procedure extends nullity notions functorially.

Nullity concepts are unified across different mathematical contexts.

Abstract

The notions of null-sets and nullity are present in all discourses of mathematics. They are based on the dual-pair of notions of "almost-every" and "almost none". A notion of nullity corresponds to a choice of subsets that one interprets as null or empty. The rationale behind this choice depends on the context, such as Topology or Measure theory. One also expects that the morphisms or transformations within the contexts preserve the nullity structures. To formalize this idea a generalized notion of nullity is presented as a functor between categories. A constructive procedure is presented by which an existing notion of nullity can be extended functorially to categories with richer structure. Nullity is thus presented as an arbitrary construct, which can be extended to broader contexts using well defined rules. These rules are succinctly expressed by right and left Kan extensions.