Categorification of Negative Information using Enrichment

作     者：Censi, Andrea Frazzoli, Emilio Lorand, Jonathan Zardini, Gioele 

作者机构：Institute for Dynamic Systems and Control Department of Mechanical and Process Engineering ETH Zurich Switzerland 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2022年

核心收录：

摘      要：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 → norphism. Norphisms do not compose by themselves;rather, they use morphisms as catalysts. After providing several applied examples, we connect nategories to enriched categtory 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. © 2022, CC BY.