The interplay between Grothendieck topoi and logic

Abstract

Grothendieck topoi are generalizations of the categories of sheaves over topological spaces (familiar from algebraic geometry). The structure of a Grothendieck topos permits an interpretation of a kind of first-order logic called geometric logic. We explore the interplay between these two viewpoints, guided by the 1987 article On Representations of Grothendieck Toposes by M. Barr and M. Makkai, and the 1977 book First Order Categorical Logic by G.E. Reyes and M. Makkai.