INTRODUCTION

This monograph is a revised version of the author's doctoral disserta-

tion, defended at the State University of New York at Buffalo in August,

1981. The main bulk of results was obtained in the spring of 1980. The

results given in chapters 1-3 were presented by the author at a meeting

of the New York Topos Seminar at Columbia University in New York City in

December, 1980, and at the Cambridge Summer Meeting in Category Theory at

the University of Cambridge, England, in July, 1981. Some results given

in chapter 4 were presented by the author at the New Mexico Research Con-

ference on Constructive Mathematics, held at Las Cruces in August, 1980.

The core of this work (section 1.1.) is a general method of forcing

over categories which subsumes, on one hand, all known instances of forcing

in set theory, Boolean and Heyting valued models and sheaf interpretations

for both classical and intuitionistic systems, and on the other hand, con-

structions of classifying topoi in topos theory. Moreover, the generic

object obtained is shown to have a clear cohomological meaning. We give

a unifying, general theory with many examples, both new and known (cf.

[FSe], [BSc4] for other new applications). We consider forcing over cate-

gories as a way of constructing objects by geometric approximation,

including a construction of a generic model of a geometric theory ([L4],

[Ti2], [Jol], [MR]) as its special case. When determining decisive pro-

perties of a required object, one first singles out simple geometric pro-

perties (often in A,v ,3 -fragment), thus defining a geometric theory T, .

The category C of forcing conditions is given as a small (not necessarily

full) subcategory C -* • Mod(5,T,) of the category of set-models of T, ,

where morphisms in C preserve additional structure given in the problem,

e.g. C may be reduced to a poset. Evaluations at objects of C have

both adjoints, so C ** Mod(5,5 ) . All objects of C are finitely pre-

sented models ([MR] appendix, [GU]). Further geometric properties of the

required object (V,3 -properties) give a Grothendieck topology on C09

vi