Just as the notion of an elementary topos is an axiomatization of basic category-theoretic properties of the category of sets, the notion of a “category with class structure” or “category of classes” is an axiomatization of the basic category theoretic properties of the category of classes.
In contrast to the situation for elementary toposes, however, there is no unique such axiomatization with a privileged status in the literature; the field of algebraic set theory includes many variations. Here we describe the first such axiomatization due to Joyal-Moerdijk.
We work in a category that is assumed to be a Heyting pretopos with a natural numbers object. Following Joyal-Moerdijk, we have the following definition.
A class of open maps (with collection) in is a class of morphisms in that satisfies the following properties:
A class of small maps in is a class of open maps in such that every map in is exponentiable and there is a universal map in with the following property: for any we can base change along some epimorphism such that the resulting morphism is a base change of along some morphism in . Elements of are known as small maps. An object of is small if the map is small.
Intuitively, small maps are maps for which preimages of any element of are sets as opposed to proper classes.
In any Heyting pretopos the class of exponentiable maps satisfies all the axioms of a class of open maps with a possible exception of the collection axiom.
The collection axiom can be reformulated by saying that the small powerset functor preserves epimorphisms. One way to define the small powerset of is as the free -complete suplattice generated by .
A category with a class of small maps admits powerclasses if for any object there is an object with a small relation such that any object and any small relation there is a unique morphism such that is the base change of along the map . Furthermore, the internal subset relation on must be small.
A universal class is an object such that any object admits a monomorphism .
A category of classes or a class category is Heyting category with a class of small maps that admits small powerclasses and a universal class.
The full subcategory of small objects and small maps of any class category is an elementary topos.
André Joyal, Ieke Moerdijk, Algebraic set theory. Cambridge University Press, 1995. ISBN 0-521-55830-1.
Steve Awodey, An outline of algebraic set theory. PDF
Last revised on June 4, 2024 at 20:02:16. See the history of this page for a list of all contributions to it.