nLab category with class structure

Idea

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.

Definition

We work in a category $C$ that is assumed to be a Heyting pretopos with a natural numbers object. Following Joyal-Moerdijk, we have the following definition.

Definition

A class of open maps (with collection) in $C$ is a class $S$ of morphisms in $C$ that satisfies the following properties:

• $S$ contains all isomorphisms and is closed under compositions;
• $S$ is closed under base changes;
• if $g\in S$ is a base change of $f$ along an epimorphism, then $f\in S$;
• the canonical maps $0\to 1$ and $1\sqcup 1\to 1$ belong to $S$;
• $S$ is closed under binary coproducts in the category of morphisms of $C$;
• if $g=f\circ p$, where $p$ is an epimorphism and $g\in S$, then also $f\in S$;
• (collection axiom) for any epimorphism $p\colon Y\to X$ and $f\colon X\to A$ such that $f\in S$ there is an epimorphism $h\colon B\to A$, a morphism $g\colon Z\to B$ such that $g\in S$, and a morphism $w:Z\to Y$ such that $h g=f p w$ and the induced map $Z\to B\times_A X$ is an epimorphism.

Definition

A class of small maps in $C$ is a class of open maps $S$ in $C$ such that every map in $S$ is exponentiable and there is a universal map $\pi\colon E\to U$ in $S$ with the following property: for any $f\in S$ we can base change $f$ along some epimorphism $p$ such that the resulting morphism $f'$ is a base change of $\pi$ along some morphism in $C$. Elements of $S$ are known as small maps. An object $X$ of $C$ is small if the map $X\to 1$ is small.

Intuitively, small maps are maps $X\to Y$ for which preimages of any element of $Y$ 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 $X$ is as the free $S$-complete suplattice generated by $X$.

Definition

A category with a class of small maps admits powerclasses if for any object $C$ there is an object $P C$ with a small relation $\in\subset C\times P C$ such that any object $X$ and any small relation $R\subset C\times X$ there is a unique morphism $r\colon X\to P C$ such that $R\to C\times X$ is the base change of $\in\to C\times P C$ along the map $id_C \times r$. Furthermore, the internal subset relation on $P C$ must be small.

Definition

A universal class is an object $U$ such that any object $C$ admits a monomorphism $U\to C$.

Definition

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.

Related notions

• class

• proper class

• large category

• category of classes

• algebraic set theory

References

• 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