Subtoposes of the Effective Topos

TL;DR

This paper introduces a new 'sight' structure to represent local operators in the effective topos, leading to the discovery of numerous new subtoposes and insights into their relationships and semantics.

Contribution

It presents the novel concept of 'sight' for representing local operators, expanding the understanding of subtoposes in the effective topos.

Findings

Established an infinite family of new basic subtoposes.

Compared new subtoposes with Turing degrees.

Provided a realizability-like semantics for subtoposes.

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.