>>12170887Introduction
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. In-
deed, 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 equiv-
alence) for a logical theory whose models are the points of a space.
2
The prototype is Stone’s Representation Theorem for Boolean alge-
bras, which relates propositional logic to Hausdorff, totally disconnected
topology. However, it takes some work to develop the idea to its full
generality.
First, the logic is not at all ordinary classical logic.
It is
an infinitary positive logic known as geometric logic. Second, we are in
general talking about predicate theories, and for these the appropriate
notion of Lindenbaum algebra is not straightforward. It is really the
“category of sets generated by the theory”.
