stylistic:style_transfer, information:extraction
In category theory

There is another approach to universes which is historically connected with category theory. This is the idea of a Grothendieck universe. Roughly speaking, a Grothendieck universe is a set inside which all the usual operations of set theory can be performed. This version of a universe is defined to be any set for which the following axioms hold:

The most common use of a Grothendieck universe U is to take U as a replacement for the category of all sets. One says that a set S is U - small if S ∈ U, and U - large otherwise. The category U - Set of all U -small sets has as objects all U -small sets and as morphisms all functions between these sets. Both the object set and the morphism set are sets, so it becomes possible to discuss the category of "all" sets without invoking proper classes. Then it becomes possible to define other categories in terms of this new category. For example, the category of all U -small categories is the category of all categories whose object set and whose morphism set are in U. Then the usual arguments of set theory are applicable to the category of all categories, and one does not have to worry about accidentally talking about proper classes. Because Grothendieck universes are extremely large, this suffices in almost all applications.

Often when working with Grothendieck universes, mathematicians assume the Axiom of Universes: "For any set x, there exists a universe U such that x ∈ U." The point of this axiom is that any set one encounters is then U -small for some U, so any argument done in a general Grothendieck universe can be applied. This axiom is closely related to the existence of strongly inaccessible cardinals.