Event Detail

Category theory deals in a mathematically natural way with certain kinds of algebraic structures on possibly very large collections of structures, such as the category Grp of all groups and the category Top of all topological spaces, in terms of the structure preserving maps (morphisms) between such objects. From this point of view, the category Cat of all categories is itself such a structure whose morphisms are the so-called functors between categories. Grp, Top, and Cat are examples of objects in Cat. Even more, if A and B are two categories, no matter how large, there is a still larger category whose objects are all the functors from A to B, and whose morphisms are the so-called natural transformations between functors. Existing set-theoretical foundations accounts for these kinds of constructions only in terms of certain kinds of restrictions, e.g. by making a distinction between small categories and large categories in a theory of sets and classes. Nevertheless, it is plausible that the foundation of an unrestricted category theory can be established without invoking such distinctions. I shall present several criteria for such a theory and show how they can be met to a considerable extent in a strong consistent extension of NFU (Quine’s system NF with urelements). However, a full foundation is blocked in stratified systems (with or without urelements) as presently treated. The talk will go over the article Enriched stratified systems for the foundations of category theory to be found at http://math.stanford.edu/~feferman/papers/ess.pdf.