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In the definition of a category on Wikipedia, it is written that a category "consists of" a class of objects and a class of morphisms, as well as binary operations for compositions of morphisms.

What concerns me about this definition is that proper classes (such as the class of all sets or the class of all groups) are by definition not allowed to be elements of any other classes (or sets). Since the category of sets "consists of" the proper class of all sets, then if we take "consists of" to mean "contains as an element", it follows that this category must be a new type of collection that is larger than both classes and sets.

The only way I think that the category of sets could itself be a class would be if we take "consists of" to mean "contains as a subclass." However, I don't see anything on the page that clarifies this.

The following questions are related, although I don't think they address my specific issue about whether categories are larger than classes:

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justin
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  • In my set theory book, a class is actually defined just by a formula. Following that philosophy, for instance "the class of all topological spaces" should be thought of as "... is a topological space", which a given object may or may not satisfy. An object $X$ is thus an object of the category of topological spaces if "$X$ is a topological space" is satisfied. – Arthur Mar 31 '16 at 22:42
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    Talking about classes in the context of ZFC is always a bit cumbersome and the short answer to your question is "don't worry about it". If that doesn't satisfy you, there are several options to formalize big categories. You could take a set theory that includes classes as your background theory (e.g. NBG or MK), you could stay within ZFC and only talk about virtual classes (i.e. every class is of the form ${x \mid \phi(x)}$ for some formula $\phi$. – Stefan Mesken Mar 31 '16 at 22:46
  • You could also ditch set theory as your foundation altogether and develop your framework using topos theory instead. These approaches have their differences, but many results carry over in a straightforward way. So - to most mathematicians - it really doesn't matter which one to take. – Stefan Mesken Mar 31 '16 at 23:05
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    You can also consider unverses. Basially, one defines and fixes a universe $U$ as a set whose elements $u$ (sets!) satisfy the usual properties except that one replaces ${x \vert \phi(x)}$ with ${x\vert \phi(x); x\in u}$. Any set $w$ which is a member of $U$ is called $small$; otherwise, $w$ is $large$. A proper class is then any $w$ that is not small. – Matematleta Mar 31 '16 at 23:14
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    @Chilango $U$ should be something resembling a Grothendieck universe? – Stefan Mesken Mar 31 '16 at 23:17
  • @Stefan: as I understand it, yes although I am by no means an expert on these. – Matematleta Mar 31 '16 at 23:19
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    It's far from agreed that it really makes sense to start from scratch with a topos. The universes approach is best, and is probably the most commonly used among category theorists and algebraic geometers. – Kevin Carlson Apr 01 '16 at 05:18
  • This question needs to be sharpened to explain why the first question linked to is not an answer. If you know what "consists of" means, and that question answers this well, then you know it doesn't mean either "has as a subclass" or "has as a member" and it makes the question about being "bigger than a class" nonsense. – Malice Vidrine Aug 10 '19 at 04:53

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The subsequent is ugly, but it is a one way to avoid, if you want, some problems with proper classes in definition. We can define the category as some class $\mathcal{C}$ such that there exists class $\mathcal{A}$ of "arrows", for which $\mathcal{C}\subseteq\mathcal{A}\times\mathcal{A}\times\mathcal{A}$, and $(\beta,\alpha,\gamma)\in\mathcal{C}$ has a sense that arrows $\beta,\alpha$ can be composed, and $\beta\alpha=\gamma$. The corresponding axioms are obvious. Then $Ob(\mathcal{C})$ and hom-sets can be proper classes. Yet the set theory is narrow for category theory.

Alex W
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