Local categories: a new framework for partiality

Restriction categories offer a categorical framework for partiality. Restriction categories are now a well-established active field of research with a rich literature [1]. In this talk, 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.

Our main result 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. In particular, given a restriction category $𝕏$, we construct a local category $𝖫[𝕏]$ by taking the subcategory of total maps of the split restriction idempotent completion of $𝕏$ and we show that this correspondence is in fact a $2$-equivalence. Furthermore, we show that every local category carries two operators which satisfy the axioms of a partial category and, from these two operators, we construct a special family of monics that makes the category into an inclusion category. Finally, we prove that every partial and inclusion category is, in fact, a local category provided it satisfies an extra assumption that we call boundedness.

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.

Moreover, we also consider our equivalence on an important class of restriction categories: inverse categories [4]. In this case, the equivalence between inverse categories and inverse local categories is a generalization of the celebrated ESN theorem for inverse semigroups [3], and recapturing some of the constructions of DeWolf and Pronk [2].

Paper. https://arxiv.org/abs/2512.03371