Abstract

) Dedicated to Ana Alex K. Simpson LFCS, Division of Informatics, University of Edinburgh, JCMB, King's Buildings, Edinburgh, EH9 3JZ Alex.Simpson@dcs.ed.ac.uk Abstract We axiomatize a notion of "classic structure" on a regular category, isolating the essential properties of the category of classes together with its full subcategory of sets. Like the axioms for a topos, our axiomatization is very simple, but has powerful consequences. In particular, we show that our axiomatized categories provide a sound and complete class of models for Intuitionistic Zermelo-Fraenkel set theory. 1. Introduction Almost thirty years on, Lawvere and Tierney's axiomatization of (elementary) toposes [10] remains a milestone in the development of contemporary mathematics. Nowadays, toposes are defined by axioms of exceptional simplicity, which nonetheless entail an astonishing richness of categorical structure [12]. Moreover, toposes are ubiquitous in mathematics. In particular, they have arisen in ge...

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