Fibrations and calculi of fractions (Q1964138)

Given a fibration \(p :{\mathcal E}\to{\mathcal B}\) and a class \(\Sigma\) of morphisms of \(\mathcal B\), the author gives an explicit construction of the fibration freely generated by \(\mathcal E\) over \(\mathcal B\) subject to the condition that the re-indexing functors over members of \(\Sigma\) are equivalences. He also shows how this technique can be used to construct the subtoposes of the effective topos which correspond to the partial combinatory algebras of \(A\)-recursive functions, for arbitrary subsets \(A\subseteq{\mathbb N}\).