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.
"Lawvere—Tierney sheaves in Algebraic Set Theory." J. Symbolic Logic 74 (3) 861 - 890, September 2009. https://doi.org/10.2178/jsl/1245158088