The concept of null in general spaces and 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.