Fibrations and calculi of fractions

Given a fibration E --> B and a class Sigma of arrows of B, one can constru ct the free fibration (on E over B) such that all reindexing functors over elements of Sigma are equivalences. In this work I give an explicit constru ction of this, and study its properties. For example, the construction pres erves the property of being fibrewise discrete, and it commutes up to equiv alence with fibrewise exact completions. I show that mathematically interes ting situations are examples of this construction. In particular, subtopose s of the effective topos are treated. (C) 2000 Elsevier Science B.V. All ri ghts reserved. MSC: 18D30; 18B25.