Final Lift Actions Associated with Topological Functors

Abstract

For a topological functor U:E→B, the fiber U−1(b), b∈B, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b,b) on U−1(b) is cocontinuous. It is shown that every cocontinuous left action of B(b,b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b=Ω, the subobject classifier, then B(Ω,Ω) inherits both a monoidal and a cocomplete poset structure. In the case B= Sets, all cocontinuous left actions of B(Ω,Ω) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.