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LAWVERE-TIERNEY SHEAVES IN ALGEBRAIC SET S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN
 

Summary: LAWVERE-TIERNEY SHEAVES IN ALGEBRAIC SET
THEORY
S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN
Abstract. 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.
Introduction
Algebraic Set Theory provides a general framework for the study of ca-
tegory-theoretic models of set theories [17]. The fundamental objects of
interest are pairs (E, S) consisting of a category E equipped with a distin-
guished family of maps S, whose elements are referred to as small maps.
The category E is thought of as a category of classes, and S as the fam-
ily of functions between classes whose fibers are sets. The research in the
area has been following two general directions: the first is concerned with
isolating axioms for the pair (E, S) that guarantee the existence in E of a
model for a given set theory; the second is concerned with the study of

  

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

 

Collections: Mathematics