In mathematics, in the framework of one-universe foundation for category theory,[1][2] the term conglomerate is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of a Grothendieck universe.
An example of such an extension is the Tarski–Grothendieck set theory, where an infinite hierarchy of Grothendieck universes is postulated.
of all sets in NBG and in MK), then these objects (proper classes) are discarded from the consideration in the new theory ("NBG/MK+Grothendieck universe").
) this in some sense does not lead to a loss of information about objects of the old theory (NBG or MK) since its representation as a model in the new theory ("NBG/MK+Grothendieck universe") means that what can be proved in NBG/MK about its usual objects called classes (including proper classes) can be proved as well in "NBG/MK+Grothendieck universe" about its classes (i.e. about subsets of
At the same time, the new theory is not equivalent to the initial one, since some extra propositions about classes can be proved in "NBG/MK+Grothendieck universe" but not in NBG/MK.
[7]: 6 The first step, made by Mac Lane,[1]: 195 [2]: 23 is to apply the term "class" only to subsets of
Mac Lane does not redefine existing set-theoretic terms; rather, he works in a set theory without classes (ZFC, not NBG/MK), calls members of