Polarities, voltages, and capacitors: a categorical approach to hulls, envelopes, and completions

This article provides a general framework in the context of category theory where one can recognize as particular instances of the same abstract construction several notions of completion, envelope, and hull, such as the completion of a Boolean algebra, the Dedekind–MacNeille completion of an ordered set, the multiplier ring of a ring, and the multiplier algebra of a C*-algebra.

Towards our goal, we lay the foundations of polarized category theory, which is a refinement of classical category theory where categories are endowed with two distinguished classes of positive and negative arrows. We define in this context the notions of polarity and voltage. We explain how a voltage can be created through a capacitor, which is essentially a polarized version of the notion of reflective subcategory. In particular, this produces a completion functor (which in the classical case is just the reflector) which assigns to each object its completion or hull. This applies even when the completion is not (and cannot be) given by a functor on the whole category, as is most often the case.

In this framework, we obtain a general theorem ensuring the existence and uniqueness of a completion functor. The corresponding completion of each object is characterized by its two universal properties with respect to positive and negative arrows.

GitHub repository: https://github.com/eulogica/polarities-voltages-and-capacitors