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In Spanier's book of algebraic topology, there is a definition of "categories" which entails "a class of objects".

I realize that the vagueness of the concept of "class of objects" is exactly used instead of "set of sets" because we want to avoid certain paradoxes of set theory.

Still, I am wondering, is there a more formal definition, or axiomatization, of what a "class of objects" mean in the concept of a category?

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    $\begingroup$ Your book lists a footnote to check "Naive Set Theory" by Halmos for reference. It avoids using the term class altogether by defining sets in such a way that they aren't needed. pg. 11 "A precise explanation of what classes really are and how they are used is irrelevant in the present approach." $\endgroup$ – muaddib Jul 20 '15 at 15:54
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    $\begingroup$ Very related questions: axiomatic definition of a category; definition of a category; large cardinals in category theory foundations $\endgroup$ – 6005 Jul 20 '15 at 17:13
  • $\begingroup$ @muaddib, are you referring to the footnote on page on page 1 of the introduction? The page introducing categories (pp. 14) doesn't have a footnote. It is nice if Halmos can avoid classes, but as long as one uses classes for categories, we need classes and thus more than Halmos, right? $\endgroup$ – PPR Jul 20 '15 at 19:51
  • $\begingroup$ @PPR - That's the one. Since your author was vague, the footnote is a clue to what he was thinking. See the answer below by Chilango. It uses naive set theory, perhaps that is the one your author intends you to use. $\endgroup$ – muaddib Jul 20 '15 at 22:05
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The notion of a class is defined rigorously in Von Neumann–Bernays–Gödel set theory, which is a conservative extension of ZFC.

Basically, you can form classes of sets using unrestricted comprehension, and you can freely take subclasses, images of classes under functions, and use the Axiom of Choice on classes of sets. However, no proper class is allowed to be an element of anything -- if a class $C$ is an element of a class $D$, then $C$ must be a set. In particular, there is no class of all classes, although there is a class of all sets.

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    $\begingroup$ Actually, comprehension is somewhat restricted in NBG, at least compared to MK. $\endgroup$ – Zhen Lin Jul 20 '15 at 16:09
  • $\begingroup$ "No class is allowed to be an element of anything", even if that thing isn't "larger". $\hspace{1.43 in}$ $\endgroup$ – user57159 Jul 21 '15 at 5:52
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It is just that, a class. That is: Given $A$, it is either true or false that $A\in\operatorname{Obj}(\mathcal C)$. Put differently, we may assume that "is an object of category $\mathcal C$" is a valid predicate.

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One way to approach this is to start with $\in $, and build up a "universe" $U$, using a standard naive set theoretic construction. $U$ is defined in such a way so as to ensure that all the usual operations of set theory applied to elements of $U$, always produce elements of $U$.

Now, we say that $u$ is $\textit {small}\Leftrightarrow u\in U$. Otherwise $u$ is said to be $\textit {large}$. Note that there are large sets: $U$ itself is large because of course $U\notin U$.

Once $U$ is defined, then you get most of the properties you want in order to do "ordinary" mathematics, such as $f:u\to v$ is small whenever $u$ and $v$ are small.

A $\textit {class}$ is defined to be any subset $S\subset U$. Note that since, by construction of $U$, $x\in u\in U\Rightarrow x\in U$, every element of $U$ is a subset of $U$ so every small set is a class. The classes that do not belong to $U$ are called $\textit {proper classes}$, $U$ being an example of a proper class.

From here, one defines Cat, the category of small categories, Cat', the category of large categories, etc.

This approach has some disadvantages. For example, if you define Cls to be the category of all classes then the set of objects of Cls is $\mathcal P(U)$ which is not a class, since its cardinality is strictly greater than that of $U$.

There are more sophisticated approaches to defining categories, but I am not expert enough to elaborate on them.

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  • $\begingroup$ Thanks for your answer. Do you have a reference for the approach you've outlined? $\endgroup$ – PPR Jul 21 '15 at 7:52
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    $\begingroup$ @PPR: William Lawvere's Elementary Theory of the Category of Sets (ETCS), or Saunders Mac Lane's Categories for the Working Mathematician might bt good starts. $\endgroup$ – Matematleta Jul 21 '15 at 14:10

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