Lawvere-Tierney Sheaves in Algebraic Set Theory

Journal of Symbolic Logic 74 (3):861 - 890 (2009)
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
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    Nathanael Leedom Ackerman (2010). Relativized Grothendieck topoi. Annals of Pure and Applied Logic 161 (10):1299-1312.
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