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

Topological incidence rings and the incidence functors