Australian Category Seminar

The fundamental theorem of restriction categories establishes that every restriction category embeds into a category of partial maps, which is a split restriction category constructed from an $\mathcal{M}$-category. In this talk, we prove a stronger version of this result. We show that restriction categories are equivalent to local categories, which are equivalent to categories equipped with a special system of monics that generalizes $\mathcal{M}$-categories. We present three new frameworks for partiality: local categories, in which partiality is assigned to objects rather than morphisms, partial categories, in which partiality is encoded by two operators on morphisms, and categories with inclusion systems, in which partiality is encoded by a choice of "inclusion" maps.