Lawvere-Tierney sheaves in algebraic set theory

TL;DR

This paper develops a general framework for defining sheaves within Algebraic Set Theory using Lawvere-Tierney coverages, broadening the scope beyond traditional topos-theoretic approaches.

Contribution

It introduces a novel approach to internal sheaves in Algebraic Set Theory based on Lawvere-Tierney coverages, relaxing axioms for small maps.

Findings

Unified framework for sheaves in Algebraic Set Theory

Broader applicability beyond topos theory

Subsume existing topos-theoretic results

Abstract

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.