# Formal Definition of Large and Small Categories

A category $\mathbf{C}$ consists of:

1. A class/collection/set $ob(\mathbf{C})$ of objects.
2. A class/collection/set $hom(\mathbf{C})$ of morphisms.
3. ... (I understand the remaining part of the definition).

My questions are:

1. Since the definition of class is informal (below), what is the class/collection/set of objects/morphisms. Are they grouped into sets/classes/collections or if it's just generic.
2. This relates to large and small categories. Wondering what a proper class is. If it's just "entities that are not members of another entity".
3. Is $\mathbb{Z}$ or $\mathbb{R}$ a small or large category (as an example).
4. What is a class, which is used to define a large category.

The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.

Outside set theory, the word "class" is sometimes used synonymously with "set".

• In ZFC, a class isn't informal, it's meta-logical. Each first-order formula in ZFC with one free variable gives rise to a class. For example, $\varphi(S)\equiv\exists x.\forall y.y\in S\Leftrightarrow y=x$ specifies the class of all singleton sets. $\varphi$ would be called a proper class if we could show $\neg\exists Z.\forall x.x\in Z\Leftrightarrow\varphi(x)$ which we can for the example I chose. Specifically, if such a $Z$ existed then $\bigcup Z$ would be the universal set to which we can apply Russell's paradox to get a contradiction. – Derek Elkins May 4 '18 at 1:42
• While you're editing, I don't understand your first question at all. What is the relevance of the fact that a class is informal (though see Derek's comment on why you may be reading too much into that)? Can we dispense with the slashes and say we have a class of objects and a class of morphisms? (It's unclear what role they have in your confusion). For 3, in what sense are you considering $\mathbb Z$ or $\mathbb R$ a category? What are the objects? What are the morphisms? If the class of objects in some category is $\mathbb Z$ or $\mathbb R,$ then, since those are sets, it is a small category. – spaceisdarkgreen May 4 '18 at 2:23
• True, removed the class part since I looked up other defs of categories and they used "collection" and stuff. A class has a specific meaning in some cases, I would prefer to say it's a collection (so it's a set or a bag/multiset). Trying to define it formally. $\mathbb{Z}$ I am just familiar with, so wanted to see if that was a category. But the problem is... "since those are sets" I don't see why not call them classes then :). Large/small categories is defined by class vs. set. What is a class (used in large category). – Lance Pollard May 4 '18 at 2:29

Elementary category theory (more or less) needs two set-theoretic "universes" as input, one contained in the other. Members of the smaller universe are called small sets, and those of the larger universe are called large sets.

While the universe of small sets needs to be fairly well behaved, elementary category theory doesn't ask much about the large universe, and is mostly independent of the technical details. So to be compatible with a wider variety of preferences on foundations, definitions are given in a way that avoid fixing these details.

While it's fairly inconvenient for category theory, a fairly common preference for mathematical foundations is first-order ZFC. In this context, the two universes you use for founding category theory are as follows.

• The small universe consists of sets
• The large universe consists of the (first-order) unary predicates (with parameters) you can express in the language of ZFC

In the traditional set theoretic context, one doesn't use the large/small language. Instead, one usually uses the term class for the members of the large universe. A proper class is a class that is not a set.

Note that the members of the large universe aren't members of the domain of discourse, so reasoning about them is delicate. There's no problem working with individual classes, but reasoning about them in general is a tricky proposition, requiring the use of schemas or going up to second order logic.

Typical examples of a proper class are the class of all sets, the class of all ordinals, the class of all groups, and such.

An example of more convenient foundations is to simply ask for there to be a set of small sets and a set of large sets, both of which are Grothendieck universes. More generally, you might even ask for Tarski's axiom of universes.

• If a proper class is not a set, wondering how saying "the class of all x" is not a set. It seems the all makes it a collection of objects. Thank you for this detailed description. – Lance Pollard May 4 '18 at 6:29
• @LancePollard "The class of all x" is a class which is not a set. For a class, roughly speaking, all you need is a defining formula of its objects (which you have given in this instance). But this collection (class) need not be a set. And in this instance (in ZFC) it provably isn't. – Stefan Mesken May 4 '18 at 12:30