Relative compactness conditions for topos

Abstract

In this paper a systematic study is made of various notions of proper map in the context of toposes Modulo some separation conditions a proper map Y X of spaces is generally understood to be a continuous function which preserves compactness of subspaces under inverse image and which therefore in particular has compact bers In this spirit a rst denition of proper map between toposes was put forward by Johnstone in There a map of toposes fF E was called proper if fF is a compact lattice object in the topos E This is probably the most direct way of expressing that F is compact when viewed as a topos over the base E In fact Johnstone used the term perfect rather than proper and developed the theory mostly in the context of localic maps between toposes see