Categorification of Negative Information using Enrichment

TL;DR

This paper introduces a novel categorical framework called nategories to formalize negative information in engineering reasoning, connecting it to enriched category theory and enriching the understanding of negative and positive information.

Contribution

It proposes the concept of nategories with norphisms to model negative information and links this to enriched category theory, providing a formal categorical foundation for negative information.

Findings

Norphisms model negative information as opposed to positive morphisms.

Nategories incorporate norphisms and morphisms with specific composition rules.

Categories enriched in de Paiva's dialectica categories define nategories with regularity properties.

Abstract

In many engineering applications it is useful to reason about "negative information". For example, in planning problems, providing an optimal solution is the same as giving a feasible solution (the "positive" information) together with a proof of the fact that there cannot be feasible solutions better than the one given (the "negative" information). We model negative information by introducing the concept of "norphisms", as opposed to the positive information of morphisms. A "nategory" is a category that has "nom"-sets in addition to hom-sets, and specifies the interaction between norphisms and morphisms. In particular, we have composition rules of the form morphism + norphism $\to$ norphism. Norphisms do not compose by themselves; rather, they use morphisms as catalysts. After providing several applied examples, we connect nategories to enriched category theory. Specifically, we prove that categories enriched in de Paiva's dialectica categories GC, in the case C = Set and equipped with a modified monoidal product, define nategories which satisfy additional regularity properties. This formalizes negative information categorically in a way that makes negative and positive morphisms equal citizens.