Topological incidence rings and the incidence functors

Abstract

We focus on working on incidence rings, a class of (possibly infinite) matrix rings indexed by ordered sets. Some general properties about them are given, including how they are always the inverse limit of finite matrix rings, giving a natural way to define their topology. We then show that such construction can be translated as a functor from the category of locally finite, preordered sets to the category of (topological) incidence rings that maps colimits to limits. We also shortly discuss the consequences of these results to the class of groups of units of incidence rings, as the unit functor allow us to translate most of the results.