Localic Topoi

Among the Grothendieck topoi those of the form Sh(S) for some topological space S play a special (and motivating) role. In this chapter we consider a related class of topoi those of the sheaves on a so-called “locale”. In the case of a topological space S, a sheaf is a suitable functor on the lattice O(S) of open sets of S, where the lattice order is defined by the inclusion relation between open sets. Thus the notion of a sheaf can be explained just in terms of the open sets of S, without any use of its points. Any suitable such lattice (one which is complete, with an infinite distributive law) may be taken as defining a modified sort of topological space, a so-called “locale”. The beginning sections of this chapter provide an introduction to the study of such locales, motivated by the topological examples. It will turn out that a topological space is essentially determined by its lattice of open sets when that space S has the property of “sobriety”, but, beyond that point, spaces and locales diverge.