The MacLane Problem on the Set-Theoretic Foundation for Category Theory. Ii

Abstract

The crisis that arose in naive set theory at the beginning of 20th century led to the appearance of several precise axiomatic constructions of theories of mathematical totalities. Among them, set theory in Zermelo–Frankel’s axiomatics (ZF) (see [1]) and the class theory and set theory in the axiomatics of Neumann–Bernays–Goedel (NBG) (see [2] and [3]) are the most widely used. These axiomatic theories remove all known paradoxes of naive set theory at the expense of a sharp restriction of possible representational tools. At the same time, they give the possibility of involving almost all mathematical objects and constructions existing at that time in the framework of these theories. However, even in 1945, in the fundamental work [4], Eilenberg and MacLane introduced a new mathematical concept of a category. Since that time, category theory has become an independent field of mathematics. However, from the very beginning, category theory encoutered the unpleasant fact that it cannot be included in the framework of the axiomatic set theory ZF and that of the axiomatic theory of classes and sets NBG (see [4]). Therefore, MacLane in [5], written in 1959, has posed the general problem of constructing a new and more flexible axiomatic set theory which can serve as an adequate logical foundation of the whole naive category theory. Variants of new axiomatic theories of mathematical totalities adopted for one or another need of category theory were suggested by Ehresmann [6], Dedecker [7], Sonner [8], Grotendieck [9], da Costa [10] and [11], Isbell [12], MacLane [13] and [14], Fefferman [15], Herrlich and Strecker [16], and others. The axiomatics of Ehresmann–Dedecker and Sonner–Grotendieck contains a very strong axiom in addition to the axioms of ZF, which asserts that every set is contained in a certain universal set. In the axiomatics of MacLane, it is assumed that there exists at least one universal set. Versions close to them were suggested by Isbell and Feferman. The axiomatics of Herlich–Strecker deals with objects of the following three types: sets, classes, and conglomerates. Within the framework of each of these axiomatic theories, some definitions of a category and a functor were given. However, the concept of a category given in [12–16] is not closed under important operations of naive set theory such as the “category of categories” and the “category of functors” (see [17, 8.4]). Within the framework of the axiomatic theories in [6–9], the definitions of a U -category and a U functor which consist of elements of a universal set U were given. Such a concept of a category is closed under operations such as the V-category of U-categories and the V-category of U -functors, where V is a certain universal set containing the universal set U . However, U -categories can contain objects of a bounded cardinality, which is not natural from the point of view of naive category theory. Therefore, in [13], MacLane noted the following about this approach: “This stronger assumption evidently provides for each universe U a category of all those groups which are members of U . However, this does not provide any category of all groups.” The second axiomatics of da Costa in [10] and [11] has the same shortcoming. His first axiomatics in [10] and [11] is free from this shortcoming, and, at the same time, it is closed with respect to the operation of naive category theory mentioned above. Therefore, it is the most preferable among all the