Editing Category of sets (section)

==Foundations for the category of sets==

In [[Zermelo–Fraenkel set theory]] the collection of all sets is not a set; this follows from the [[axiom of foundation]]. One refers to collections that are not sets as [[proper class]]es. One cannot handle proper classes as one handles sets; in particular, one cannot write that those proper classes belong to a collection (either a set or a proper class). This is a problem because it means that the category of sets cannot be formalized straightforwardly in this setting. Categories like '''Set''' whose collection of objects forms a proper class are known as [[Large category|large categories]], to distinguish them from the small categories whose objects form a set. 

One way to resolve the problem is to work in a system that gives formal status to proper classes, such as [[NBG set theory]]. In this setting, categories formed from sets are said to be ''small'' and those (like '''Set''') that are formed from proper classes are said to be ''large''.

Another solution is to assume the existence of [[Grothendieck universe]]s. Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for instance if a set belongs to a universe, its elements and its powerset will belong to the universe).  The existence of Grothendieck universes (other than the empty set and the set <math>V_\omega</math> of all [[hereditarily finite set]]s) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence of [[strongly inaccessible cardinal]]s. Assuming this extra axiom, one can limit the objects of '''Set''' to the elements of a particular universe. (There is no "set of all sets" within the model, but one can still reason about the class ''U'' of all inner sets, i.e., elements of ''U''.)

In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes. (This is necessarily a [[proper class]], but each Grothendieck universe is a set because it is an element of some larger Grothendieck universe.) However, one does not work directly with the "category of all sets". Instead, theorems are expressed in terms of the category '''Set'''<sub>''U''</sub> whose objects are the elements of a sufficiently large Grothendieck universe ''U'', and are then shown not to depend on the particular choice of ''U''. As a foundation for [[category theory]], this approach is well matched to a system like [[Tarski–Grothendieck set theory]] in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all '''Set'''<sub>''U''</sub> but not of '''Set'''.

Various other solutions, and variations on the above, have been proposed.<ref>{{harvnb|Mac Lane|1969}}</ref><ref>{{harvnb|Feferman|1969}}</ref><ref>{{harvnb|Blass|1984}}</ref>

The same issues arise with other concrete categories, such as the [[category of groups]] or the [[category of topological spaces]].