A classifying localic category for locally compact locales with application to the Axiom of Infinity (poster)

For an internal category C in a cartesian category C we define, naturally in objects X of C, PrinC(X). This is a category whose objects are principal c C-bundles over X and whose morphisms are principal c(C↑)-bundles. Here c(₎ denotes taking the core groupoid of a category (same objects but only isomorphisms as morphisms) and C↑ is the arrow category of C (objects morphisms, morphisms commuting squares). We show that X ↦ PrinC(X) is a stack of categories and call stacks of this sort lax-geometric. We then provide two sufficient conditions for a stack to be lax-geometric and use them to prove that the pseudo-functor X ↦ LKSh(X) on the category of locales Loc is a lax-geometric stack. Here LKSh(X) is the category of locally compact locales in the topos of sheaves over X, $Sh(X)$. Therefore there exists a localic category CLK such that LKSh(X) Prin_CLK(X) naturally for every locale X. We then show how this can be used to give a new localic characterisation of the Axiom of Infinity.