"universe" in set theory and category theory

In all applications of the theory of sets, all sets under investigation take place in the context of the universal set $U$. What exactly is the purpose of the universal set in set theory? $U$ is infinite, why do we need to mention this infinite context in regards to set theory?

In category theory is there such a notion of a universe? Would it be the category of categories? I have a feeling that the answer has to do with Russell's paradox, but if you could elaborate that'd be great, I'm a beginner.

Also, is there such a thing as "outside" of a universe?

Solution 1:

It is convenient to know that if you have a bunch of sets and then do something with these sets (e.g., take their cartesian product), then the object you will get will still be manageable. In other words, we want to know that we can perform lots of convenient operations with sets we like and still remain in the same 'universe' of sets we like. Russele's Paradox demonstrates clearly that this is nothing trivial.

In set theory the universe of sets, or set of discourse, is used in two different ways. Naively, it just specifies a set where all the elements come from when you write $\forall$ or $\exists$. For instance, when talking about real numbers it is usually assumed the set of discourse is $\mathbb R$. In axiomatic set theory one specifies some axioms of set theory and then consideres models of the axioms. A model of the axiom is itself a set whose elements are the sets that the model defines. So, 'set' here is used in two totally different ways. The universe is the set $M$ which is the model of the axioms of set theory. The elements of $M$ are all of the sets that the model allows. The set $M$ itself is, typically, not an element on $M$, thus is not a set in the model.

In category theory it is convenient to know that given a bunch of categories we can perform lots of constructions with them, like forming categories of functors. This means that the underlying set theory we employ should be strong enough to allow for lots of 'big' constructions. Grothendieck introduced the notion of a tower of universes of sets to manage these constructions. This is required since typical categories are big in the sense that their objects do not form a set of the same magnitude as the hom-sets do. For instance, for any two groups $G,H$ the hom-set of all group homomorphisms $\psi:G\to H$ is indeed a set (even if the groups are very big). So, when we define the category $Grp$, each hom-set is just a set, but the class of all groups does not form a set. There is thus a hierarchy, using Grothendieck universes, of sizes for categories and various constructions may transcend the size of the categories it operates on, or it may not, depending on the constructions.

On that note, the category $Cat$ usually refers to the category of all small categories, meaning all categories whose objects form a set (in a fixed level of the tower of universes which is ambient and assumed fixed). Thus, the category $Cat$ itself is not a small category (avoiding such silly paradoxes as the category of all categories containing itself as an object). Instead, $Cat$ is a larger category than any of the categories it contains (which makes sense). $Cat$ is still a manageable object using universes, however, the category $CAT$ of all categories is huge and presents many more difficulties if one really needs to deal with it.

To see the importance of size issues in category theory, recall that a category is called small complete if it has all set-indexed limits, and that a category is a poset if between any two of its objects there is at most one morphism. There are plenty of small complete categories (e.g., $Set$, $Grp$) which are clearly not posets. Interestingly, if one looks at complete categories (i.e., those having all limits, with no size restriction), then any such category must be a poset.

Solution 2:

What is true is that most mathematical discussion takes places in the context of some set, which is called "the universe of discourse." You can't have a "universal set;" Russel's paradox dispatches that notion.

Solution 3:

In "real" set theory, there is usually no universal set. The reason is that we care about the collection of all sets, and in modern set theory this is not a set. It's a collection which we can define, and we can discuss meaningfully, but it's not a set.

On the other hand, in almost any mathematical application, in particular in logic where we have structures for a language and models of a theory, the universe of discourse is a set. So what we do is we work with respect to that set, rather than the whole universe of set theory.

In category theory there is a problem with the category of categories, and that's the same issue which makes the set of all sets problematic. But there are ways to overcome this issue.

The term universe in category theory, often mean some set theoretical universe which we can see as a set from the point of view of a larger universe, then we can work inside that universe and outside that universe. This allows us to talk about categories whose objects and morphisms are members of that universe, and categories which are too large with respect to the first universe, but are small enough with respect to the second universe.

Solution 4:

This is an interesting question and, even though it was asked a year ago, will give my answer.

Actually, the question sounds to be about many notions with the same or close name:

• Universal set with two meanings: (a) a set, about the subsets of which we discuss, like in case of a Boolean algebra of sets, (b) a set, about the elements of which we discuss, like in the book Elementary "Set Theory with a Universal Set" by Holmes Randall (1998)
• Universe of discourse of a theory $T$ - a collection, which together with some relationships and operations form a model of a theory. In particular, $T$ can be one of many axiomatic set theories. Thus, the notion of "universe" is not used only in set theory.
• Universe as a special kind of set - a set which contains the elements of its elements, and has other properties like it was described by Ittay Weiss above.

Each of these notions manifest differently "in applications". The "rule of thumb" to distinguish between "set" and "universe" is that a universe is a special kind of set.