Basic Subtoposes of the Effective Topos

Abstract

We study the lattice of local operators in Hyland’s Effective Topos. We show that this lattice is a free completion under internal sups indexed by the natural numbers object, generated by what we call basic local operators. We produce many new local operators and we employ a new concept, sight, in order to analyze these. We show that a local operator identified by A.M. Pitts in his thesis, gives a subtopos with classical arithmetic.