Prologue

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”. These different aspects arose independently around 1963: with A. Grothendieck in his reformulation of sheaf theory for algebraic geometry, with F. W. Lawvere in his search for an axiomatization of the category of sets and that of “variable” sets, and with Paul Cohen in the use of forcing to construct new models of Zermelo-Frwnkel set theory.