Lawvere–Tierney sheaves in Algebraic Set Theory

We present a solution to the problem of defining a counter- part 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 exist- ing topos-theoretic results.

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