Large Categories
Table of Contents
1. Overview
The basic problem is, when defining a category, we have some "collection" of objects and some "collection" of morphisms. But there are two natural notions of "collection" among set theorists: sets and proper classes.
A category is called [Strictly] Large if it consists of a proper class of objects and a proper class of morphisms.
A category is called Locally Small if it consists of a proper class of objects, and for any two objects \(X\) and \(Y\) the hom-set \(\hom(X,Y)\) is a proper set.
1.1. Foundation Dependence
The real crux of the problem is that, determining if a category is small or large will depend on your axioms of set theory. This is why Tarski–Grothendieck set theory was developed, to "iterate" the NBG construction endlessly.
If we work with Morse–Kelley set theory, then we have some problems:
- The category of functors between two strictly large categories would have a collection of classes for its objects, which is beyond MK.
See Andreas Blass's answer to this question on Mathoverflow.
2. Small Categories
A category is called Small if it consists of a proper set of objects and, for any two objects \(X\) and \(Y\), the hom-set \(\hom(X,Y)\) is a proper set.
3. References
- Paul Blain Levy,
"Formulating Categorical Concepts using Classes".
arXiv:1801.08528, 21 pages - Mike Shulman,
"Set theory for category theory".
arXiv:0810.1279 - Jiri Adamek, Horst Herrlich, and George Strecker,
Abstract and Concrete Categories: The Joy of Cats. Eprint: PDF, 524 pages, part 0, chapter 2.