Modified realizability subtoposes and total Weihrauch reducibility

Abstract

In recent years, there has been rapid development in the foundational study of oracle computability from the perspective of Lawvere-Tierney topologies and their sheaves. In this article, we formulate and analyze the notion of reducibility within the framework of total computability. Then, using sheaf subtoposes derived from oracles in the total computable setting, we establish separations between various hierarchies of logical principles, including the hierarchies of the weak law of excluded middle $\mathbf{WLEM}$, the lessor limited principle of omniscience $\mathbf{LLPO}$, and Markov's principle $\mathbf{MP}$.