 
Summary: LAWVERETIERNEY 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 LawvereTierney 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 topostheoretic results.
Introduction
Algebraic Set Theory provides a general framework for the study of ca
tegorytheoretic 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
