The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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 (2)

Comment #1481 by verisimilidude on

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 on

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


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