Topological incidence rings and the incidence functors
João V. P. e Silva
TL;DR
This work provides a categorical framework for topological incidence rings by constructing a contravariant incidence functor from the category of locally finite prosets to topological rings, with colimits in prosets corresponding to limits in rings. It shows that incidence rings over a ring $P$ are inverse limits of finite matrix rings, endowing them with a natural Hausdorff topology, and extends the construction to topological groups via the unit functor when $P$ is commutative. The theory yields structural results: the unit group $ extup{GL}_oldsymbol{ extΛ}(P)$ is the inverse limit of finite $ extup{GL}_oldsymbol{ extα}(P)$, and solvability properties of these groups can be inferred from finite-block behavior; in certain non-poset preorders, the unit groups fail to be solvable or prosolvable. Overall, the paper unifies incidence algebras and their unit groups under a cohesive, limit-preserving categorical perspective, enabling generation from finite components and providing tools to study solvability in topological settings.
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.