nLab class function (set theory)

This entry is about the term of “class function” as it is used in set-theoretical contexts. For the notion of the same name in algebra see at class function.

Idea

Recall that in set theory a function $f$ from a set $S_1$ to a set $S_2$ may be encoded in terms of a relation on the Cartesian product $S_1 \times S_2$ of the two sets, namely the subset $R \subset S^1 \times S^2$ with $R = \{ (x,y) \vert y = f(x)\}$ (the graph of the function).

This concept has an evident generalization to the case where $S_1$ and $S_2$ are allowed to be proper classes. In this case one speaks of class functions.

Definition

A class function is a class $R$ which is a relation with the property that if $(x,y)\in R$ and $(x,y')\in R$, then $y=y'$.

A class function need not be a set.

Class functions are an important concept when formalizing category theory on set-theoretic foundations without using universes, especially when

• constructing functors step by step, e.g. by first constructing an assignment between two classes of objects, and then compatibly augmenting this assignment to make it a functor,

• most especially when constructing functors on functor categories, whose classes of objects tend to be proper classes.

Example in Category Theory

An example is Mac Lane’s discussion p. 23 of the category of all classes in his chapter on Foundations. There, the class of arrows is defined by writing “its arrows [are] all functions $f\colon C\rightarrow C'$ between classes”.

Class functions are the morphisms in a category with class structure.

Example in Set Theory

Arguably the prototypical example of an essential use of the concept of class functions in set theory is Easton's theorem Theorem 15.18. on the behaviour of the class function $x\mapsto 2^x$ on the class of all cardinals. (In Easton’s theorem, the domain of the class function in question is the proper class of all regular cardinals.)

Often, authors resort to synonyms like operation or assignment when writing about proper classes, with p. 23 being a counterexample.

References

• Saunders Mac Lane. Categories for the Working Mathematician Second Edition. Springer (1998)

• T. Jech. Set theory. Third Edition. Springer (2002)