Local categories: a new framework for partiality

TL;DR

This paper introduces three new categorical frameworks for modeling partiality, establishing their equivalences and translating key concepts, thereby enriching the theoretical understanding of partial structures in category theory.

Contribution

It presents local, partial, and inclusion categories as new frameworks for partiality, proving their equivalences to existing restriction categories and extending key concepts like inverse categories.

Findings

Restriction categories are 2-equivalent to local categories.

Partial categories are 2-equivalent to inclusion categories.

Inverse categories are 2-equivalent to inverse local categories.

Abstract

Restriction categories provide a categorical framework for partiality. In this paper, we introduce three new categorical theories for partiality: local categories, partial categories, and inclusion categories. The objects of a local category are partially accessible resources, and morphisms are processes between these resources. In a partial category, partiality is addressed via two operators, restriction and contraction, which control the domain of definition of a morphism. Finally, an inclusion category is a category equipped with a family of monics which axiomatize the inclusions between sets. The main result of this paper shows that restriction categories are $2$-equivalent to local categories, that partial categories are $2$-equivalent to inclusion categories, and that both restriction/local categories are $2$-equivalent to bounded partial/inclusion categories. Our result offers four equivalent ways to describe partiality: on morphisms, via restriction categories; on objects, with local categories; operationally, with partial categories; and via inclusions, with inclusion categories. We also translate several key concepts from restriction category theory to the local category context, which allows us to show that various special kinds of restriction categories, such as inverse categories, are $2$-equivalent to their analogous kind of local categories. In particular, the equivalence between inverse (restriction) categories and inverse local categories is a generalization of the celebrated Ehresmann-Schein-Nambooripad theorem for inverse semigroups.