Dagger categories of relations: The equivalence of dilatory dagger categories and epi-regular independence categories

Abstract

Several categories look like categories of relations, but do not fit the established theory of relations in regular categories. They include the category of surjective multivalued functions, the category of injective partial functions, the category of finite probability spaces and stochastic matrices, and the category of Hilbert spaces and linear contractions. To explain these anomalous examples, we develop a parallel theory of relations in epi-regular independence categories. Just as regular categories correspond to tabular allegories, epi-regular independence categories correspond to dilatory dagger categories. The equivalence maps epi-regular independence categories to their associated dagger category of relations, and dilatory dagger categories to their wide subcategory of coisometries.