Topos theory provides an important setting and language for much of mathematical logic and set theory. It is well known that a typed language can be given for a topos to be regarded as a category of sets. This enables a fruitful interplay between category theory and set theory. However, one stumbling block to a logical approach to topos theory has been the treatment of geometric morphisms. This book presents a convenient and natural solution to this problem by developing the notion of a frame relative to an elementary topos. The authors show how this technique enables a logical approach to be taken to topics such as category theory relative to a topos and the relative Giraud theorem. The work is self-contained except that the authors presuppose a familiarity with basic category theory and topos theory. Logicians, set and category theorists, and computer scientist working in the field will find this work essential reading.
'The present book represents what must surely be an extreme point in the spectrum of possible approaches: the one which makes the greatest possible use of formal languages. The material, throughtout the books, is of a highly technical nature ... By providing a language adequate to handle indexed categories over a topos (and much more besides), the authors have done much to "demystify" indexed category theory and make it accessible to those for whom the
sight of a 2-categorical diagram is liable to induce instant incomprehension. All such people ... should make the effort to master the technicalities in this book, if they wish to get closer to an
understanding of what really goes on in the 2-category of toposes.'
P.T. Johnstone, Zentralblatter, 1993
Introduction; Local set theories; Partial function theory `L'; Equationals; Categories in a topos; Topoi in a topos; A representation theorem for geometric morphisms; Local set theories in S; The theory of a topos in S; Topologies and sheaves; The relative Giraud theorem; Appendices.