nLab
large category

Context

Category theory

Foundations

Contents

Idea

A large category is a category which is not (necessarily) small.

There are some variations in usage depending on the foundations chosen. Also, not all authors agree on whether a large category is not small, or merely not necessarily small (i.e., whether small categories are also large).

Definitions

The precise meaning of the above definition depends on the foundations chosen.

In all cases, it is somewhat ambiguous whether “large category” means “properly large,” i.e., large and not small, or whether small categories should be considered as a subclass of large categories. Usage may vary depending on need.

Largeness and moderateness

A moderate category may be defined as one whose collections of objects and morphisms are no bigger than the size of the universe of small sets. This is related to largeness in different ways depending on the foundations.

Properties

Many tools and results about small categories, in particular concerning limits indexed by such a category and functor categories, fail for large categories. For instance:

There are various notions and techniques to deal with this problem and reduce or relate large categories to small categories as much as possible:

References