Abstract

. There is a close relationship between category theory and logic. For example, elementary toposes have just enough properties to interpret intuitionistic higher-order logic, and we think of toposes as `categories of sets'. In fact, a topos with a natural numbers object is an adequate universe in which to develop intuitionistic mathematics, and such a topos may be seen as a categorical analogue of a model of intuitionistic Zermelo-Fraenkel set-theory. In this paper we implement the categorical analogue of Bernays-Godel set-theory. We introduce the notion of small structure on a category, and if small structure satises certain axioms we can think of the underlying category as a category of classes. Our axioms imply the existence of a co-variant powerset monad on the underlying category of classes, which sends a class to the class of its small subclasses. Simple xed points of this and related monads are shown to be models of intuitionistic Zermelo-Fraenkel set-theory (IZF). ...

Citations

379	Introduction to Higher Order Categorical Logic - Lambek, Scott - 1986
186	Non-well-founded sets - ACZEL - 1988
119	Handbook of categorical algebra - Borceux - 1994
40	Strong functors and monoidal monads - Kock - 1972
35	Algebraic Set Theory - Joyal, Moerdijk - 1995
23	Elementary axioms for categories of classes - Simpson - 1999
7	Set Theory with a Universal Set - Church - 1974
5	Cartesian monads on toposes - Johnstone - 1997
5	A categorical theory of cumulative hierarchies of sets. Comptes Rendus Mathématiques de l’Académie des Science - Joyal, Moerdijk - 1991
2	Regular categories and regular logic. Course notes - Butz - 1998