A seminar talk presented at The Antwerp Algebra Colloquium in the framework of the Master Program at the University of Antwerp.

Abstract

The goal of this talk is to introduce the audience to the theory of elementary topoi. These are categories which generalize the category of sets (and functions), not only in the sense of having the same universal constructions, but they also become models of the set theory of Zermelo-Fraenkel. By this I mean that certain morphisms in a topos form the abstract formulas in such a theory, called the internal logic, and after de fining when such formulas are valid, it can be shown that the axioms of (intuitionistic) propositional- and predicate logic and set theory are valid.

The central notion of a topos is that of a subobject classifi er. This is a particular kind of object which plays the role of a two-element set by allowing each subobject to be described by a unique morphism into that object (just as a subset is uniquely de fined by a function into a two-element set). A formula then also corresponds with a morphism into this object, so to form all formulas of (intuitionistic) propositional logic, the subobject classifi er needs to have a specifi c kind of partial order structure, that of a Heyting algebra. The fundamental theorem of topoi allows us the interpret the quanti fiers.