Section 3.3 (000A): Classes—The Stacks project

3.3 Classes

Informally we use the notion of a class. Given a formula

$\phi (x, p_1, \ldots , p_ n)$ , we call

$C = \{ x : \phi (x, p_1, \ldots , p_ n)\}$

a class. A class is easier to manipulate than the formula that defines it, but it is not strictly speaking a mathematical object. For example, if

$R$ is a ring, then we may consider the class of all

$R$ -modules (since after all we may translate the sentence “

$M$ is an

$R$ -module” into a formula in set theory, which then defines a class). A proper class is a class which is not a set.

In this way we may consider the category of

$R$ -modules, which is a “big” category—in other words, it has a proper class of objects. Similarly, we may consider the “big” category of schemes, the “big” category of rings, etc.

Comments (5)

Comment #1481 by verisimilidude on June 05, 2015 at 14:52

You say in section 3.3 (tag 000A) that "A proper class is a class which is not a set." Yet in the previous section, 3.2, the declaration was made that everything is a set. So is the set of proper classes an empty set? And if so then what is the use of defining a proper class. If this is not true then I think we readers are owed a little more explanation of the term "proper class".

Comment #1482 by Johan on June 05, 2015 at 18:34

Note the word "informally" at the beginning of the paragraph.

Comment #4157 by 蒋云江 on April 07, 2019 at 23:08

It would be nice to outline why the class of R modules is proper, or at least mention which ZFC axiom is violated.

Comment #4366 by Johan on August 29, 2019 at 19:02

Hmm. Well, given any set $S$ the set of maps from $S$ to $R$ is an $R$ -module of cardinality greater than or equal to $|S|$ . Thus we have $R$ -modules of arbitrarily large cardinality and hence these cannot all be elements of a single set.

Of course no axioms are being violated! Not sure what you meant by that statement.

I'm going to leave this alone for now unless other people speak up or make the edits themselves.

Comment #8150 by Aise Johan de Jong on February 27, 2023 at 14:46

OK starting now, I am going to delete comments on this page as it seems to attract comments which are not on the content of the Stacks project at all.

3.3 Classes

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