Fibred sets within a predicative and constructive effective topos

Abstract

We describe the fibrational structure of sets within the predicative variant $\mathbf{pEff}$ of Hyland's Effective Topos $\mathbf{Eff}$ previously introduced in Feferman's predicative theory of non-iterative fixpoints $\widehat{I{D}_1}$. Our structural analysis can be carried out in constructive and predicative variants of $\mathbf{Eff}$ within extensions of Aczel's Constructive Zermelo-Fraenkel Set Theory. All this shows that the full subcategory of discrete objects of Hyland's Effective topos $\mathbf{Eff}$ contains already a fibred predicative topos validating the formal Church's thesis, even when both are formalized in a constructive metatheory.