Subtoposes of the Effective Topos

Abstract: We seek progress in the study of subtoposes of the effective topos. First we treat Van Oosten's result that local operators on the effective topos are internally NNO-indexed joins of what we shall call 'basic' local operators. Our main innovation is the notion of a tree-like structure called 'sight', which provides a tangible representation of local operators on the effective topos. This leads in particular to the establishment of an infinity of new basic subtoposes of the effective topos. Various comparisons (inequalities and non-inequalities) in between these new examples and known examples such as Turing degrees are made. Sights also provide a realizability-like semantics for the first-order arithmetic of subtoposes of the effective topos. The text begins with an overview of relevant tripos theory and some topos-theoretic constructions of local operators.