Abstract

We study the type-theoretical analogue of Bernays–Gödel set-theory and its models in categories. We introduce the notion of small structure on a category, and if small structure satisfies 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 fixed points of this and related monads are shown to be models of intuitionistic Zermelo–Fraenkel set-theory (IZF).

Mathematical subject codes: Primary: 03E70; secondary: 03C90; 03F55

Article Outline

0. Introduction

1. Preliminaries

1.1. Regular categories

1.2. The indexed subobject functor

2. The basic axioms

2.1. Small structure

2.2. Induced small structure

2.3. The basic axioms

2.4. Further axioms

3. Representable small structure

3.1. The axiom (Representability)

3.2. Extensionality

3.3. The order on P(X)

3.4. Stability under slicing

4. Replacement, singleton, and union

4.1. Replacement

4.2. Singleton

4.3. Union and the monad structure on P

4.4. Families of small subobjects versus maps with small fibers

4.5. Local replacement

4.6. The complete characterization

5. Separation and powerset

5.1. The subobject classifier

5.2. Separation

5.3. The powerset axiom

6. A first summary

7. Intuitionistic Zermelo–Fraenkel set-theory

7.1. Simple algebras as models

7.2. The axiom (Infinity)

Acknowledgements

References