I'm not very experienced with respect to Category Theory. So if this question makes no sense I'm sorry. At any rate here is my question: If the existence or non-existence of specific sets can be independent of set theory, then how can it be that the category Set is complete under small limits?


For example, suppose you have a small category A, with objects that are linearly ordered spaces X such that: X is without smallest or largest element, X has CCC, X is complete, and X is dense in itself. And as morphisms for A, you take order preserving bijections. Now, let F be the forgetful functor from A to the underlying set.


How exactly can you define the limit over F inside of Set?


Another example, would be considering some set sized collection of Whitehead groups, call it X. Now, consider X as a category equipped with homomorphisms as morphisms. And let G be the forgetful functor from X into Set.


How exactly can one define the limit over G inside Set?


Or even better, suppose that G is the identity functor from X into Grps. Then, what happens?


Am I correct in saying that the answer depends drastically on the set theoretic universe you pick? If so, how is this not a problem with category theory?



Independence and Category Theory
Like all fields of mathematics, Category Theory is not immune to foundational questions. Although different foundations have been presented, the most common foundation for Category Theory is within Set Theory: Category Theory starts with a given universe of sets and then develops its theory. The computation of limits, colimits, and whatnot will be affected by the overlying structure of sets. However, this overlying structure of sets is not variable. When a computation depends on certain principles of Set Theory, it must be labeled as such. For example, [Grothendieck Universes](http://en.wikipedia.org/wiki/Grothendieck_universe) and [Vopenka's Principle](http://en.wikipedia.org/wiki/Vop%C4%9Bnka%27s_principle) are large cardinal axioms which have direct applications in Category Theory.


That said, Category Theory is in a unique position to deal with independence results that arise from Set Theory. Indeed, every forcing construction in Set Theory has an analogue in Category Theory via sheaves over an appropriate site. More precisely, forcing poset can be viewed as a small category and when endowed with the double-negation topology, the Grothendieck topos of sheaves over this site is equivalent to the Boolean-valued model that one obtains in via forcing. Thus independence results from Set Theory are directly visible to Category Theory by doing the computations inside a Boolean Grothendieck topos instead of the topos of sets.


