Aspects of predicative algebraic set theory, II: Realizability
One of the main goals of this paper is to give a construction of realizability models for predicative constructive set theories in a predicative metatheory. We will use the methods of algebraic set theory, in particular the results on exact completion from van den Berg and Moerdijk (2008) [5]. Thus, the principal results of our paper are concerned with the construction of an extension of a category with small maps by a category of assemblies, again equipped with a class of maps, and to show that this extension construction preserves those axioms for a class of maps necessary to produce models of the relevant set theories in the exact completion of this category of assemblies. (C) 2010 Elsevier B.V. All rights reserved.
realizability constructive set theory categorical logic