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

Abstract

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 Boolean algebra completion of a Boolean algebra, the Dedekind--MacNeille completion of an ordered set, the multiplier ring of a ring, the multiplier algebra and the von Neumann envelope of a C*-algebra. Towards our goal, we lay the foundations of \emph{polarized} category theory, which is a refinement of classical category theory where categories are endowed with two distinguished classes of \emph{positive} and \emph{negative} arrows. We define in this context the notion of \emph{polarity}, and \emph{voltage}. We explain how a voltage can be created through a \emph{capacitor}, which is essentially a polarized version of the notion of reflective subcategory. In particular, this produces a \emph{completion functor} (which in the classical case is just the reflector) which assigns to each object its completion or hull. These applies even when the completion is not (and cannot be) given by a functor on the whole category, as it is most often the case. In this framework, we obtain a general theorem ensuring the existence and uniqueness of a functorial completion functor. The corresponding completion of each object is characterized by its two universal properties with respect to positive and negative arrows.