Abstract. We give a general method of forcing over categories as a

category-theoretic universal construction 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

formal systems; and, on the other hand, constructions of classifying

topoi in topos theory (Grothendieck*s generalization of classifying spaces

considered in algebraic topology, algebraic geometry). The generic

object obtained by forcing is shown to have a clear cohomological meaning.

Furthermore, we show that iterated forcing in set theory, and Grothendieck's

construction of a lax limit of a fibred topos are the same up to Godel's

negative interpretation of classical into intuitionistic logic. This

suqgests possibilities of interapplications between logic, and algebraic

geometry and algebraic topology.

AMS 1980 Mathematics Subject Classification; 03G30, 18B25, 18D30, 03E35,

03F50.

Key words and phrases: classifying topos, geometric theory, universal

model, forcing.

Library of Congress Cataloging in Publication Data

Scedrov, Andrej, 1955-

Forcing and classifying topoi.

(Memoirs of the American Mathematical Society, ISS*T

0065-9266 ; no. 295)

"Volume 48 number 295."

Bibliography: p.

1. Toposes. 2. Categories (Mathematics) 3. Forcing

Obdel theory) I. Title. II, Series: Memoirs of the

American Mathematical Society ; no. 295.

QA3.A57 no. 295 [QA169] 510s [512\55] 83-26644

ISBN 0-8218-2294-2

IV