Presheaves over lattices: application to twisted limits of finite groups.

Abstract

The notion of “active sum of groups” appears as a special case of the notion of twisted (direct) limit of groups over partially ordered sets and lattices. This provides us with a sheaf-like theory; in fact up to introducing coverings in a lattice we obtain an extension of the theory of (pre-)sheaves over Grothendieck topologies. It is possible to introduce the point-lattice (or point-topology) and describe the transfer of structure properties from the base lattice to the point-lattice. Starting from general presheaf-action (of groups) one may then introduce a global twisted limit. Using particular functors it is then possible to define “structure sheaves” in our setting. Inspired by the theory of modules over rings, we then consider the special case of G-sets. This clears the way for application to the Burnside ring of finite groups.