Fibrations and calculi of fractions

Given a fibration and a class Σ of arrows of , one can construct the free fibration (on over ) such that all reindexing functors over elements of Σ are equivalences. In this work I give an explicit construction of this, and study its properties. For example, the construction preserves the property of being fibrewise discrete, and it commutes up to equivalence with fibrewise exact completions. I show that mathematically interesting situations are examples of this construction. In particular, subtoposes of the effective topos are treated.