Locales and Toposes as Spaces

1. Introduction Mac Lane and Moerdijk, 1992, in their thorough introduction to topos theory, start their Prologue by saying – A startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory. Indeed , a topos can be considered both as a " generalized space " and as a " generalized universe of sets ". This dual nature of topos theory is of great importance, and one can quite reasonably understand Grothendieck's name " topos " as meaning " that of which topology is the study ". Mac Lane and Moerdijk are unquestionably masters of the spatial nature of toposes, yet one could easily read through their book without grasping it. The mathematical technology is so firmly expressed in the set theory and the logic that the spatiality is obscured. The aim in this chapter is to provide a reader's guide to the spatial content of the major texts. Those texts can also provide a more detailed account of original sources and other applications than has been possible here. We have on the one hand, the logic and set theory, and, on the other, the topology. In a nutshell, the topos connection between them is that the topos acts like a " Lindenbaum algebra " (of formulae modulo equivalence) for a logical theory whose models are the points of a space.