Stability of Properties of Locales Under Groups

摘要

Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales, we show that if is a category that satisfies the axioms then so too is the category of G-objects, for any internal group G. To achieve this we prove a general categorical result: if an object S is double exponentiable in a category with finite products then so is its associated trivial G-object (S, pi (2):G x S -> S). The result holds even if S is not exponentiable. An example is given of a category that satisfies the axioms, but for which there is no elementary topos such that is the category of locales over . It is shown, in outline, how the results can be extended from groups to groupoids.