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.
AWODEY, S., GAMBINO, N., LUMSDAINE, P., WARREN, M. (2009). Lawvere-Tierney sheaves in Algebraic Set Theory. THE JOURNAL OF SYMBOLIC LOGIC, 73(3), 861-890.
Lawvere-Tierney sheaves in Algebraic Set Theory
GAMBINO, Nicola;
2009-01-01
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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