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-Gödel 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)
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