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\section*{proper class}

\hypertarget{proper_classes}{}\section*{{Proper classes}}\label{proper_classes}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{usage}{Usage}\dotfill \pageref*{usage} \linebreak
\noindent\hyperlink{category_of_classes}{Category of classes}\dotfill \pageref*{category_of_classes} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{proper class} is a [[set]] that is too big to be a set. Exactly what this means depends on the [[foundations]] of mathematics, but something must be said about it to study [[large category|large categories]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

There are several ways to deal with this and define a proper class.

A \textbf{proper class} is a [[large category|large]] [[discrete category]]. But since a large category is usually defined as having a proper class of objects, this just moves the bubble under the wallpaper to `large category' and something must be applied there.

A \textbf{proper class} is a collection that can be put in [[bijection]] with the class of all [[ordinal number|ordinals]], $Ord$. But this requires the [[global axiom of choice]] to be correct.

A \textbf{proper class} is a class that is not a [[set]]. So now we have to define `class'.

A \textbf{proper class} is a class whose cardinality is not the [[cardinal number]] of any set. This is a version of the previous definition not violating the [[principle of equivalence]]; however, in some foundations these are actually equivalent (using the [[axiom of replacement]]).

A \textbf{class} is a collection of [[sets]]. Here the bubble is moved to `collection', but we will be able to pop that bubble below. Also we might want to allow the members of a proper class to be other than sets (such as [[structured sets]]); certainly it is true, however, that a \textbf{pure class} is a collection of [[pure sets]].

A \textbf{class} is a formula in the language of [[set theory]] for a [[truth value]], equipped with a specified [[free variable]] for a set. This is a formalisation of the previous definition, but it must be interpreted metamathematically: a \textbf{formula for a class} in a given [[context]] $\Gamma$ is a formula for a truth value in the extension of $\Gamma$ by one more free variable for a set.

A \textbf{class} may even be an undefined concept; the real definition is to define a \textbf{[[set]]} as a class that is itself a member of some class. With appropriate axioms, this is equivalent to the previous definition (and conservative over set theory without classes), but it's also possible to apply stronger axioms here; this choice is the difference between $BNG$ and $MK$ as extensions of [[ZFC]].

A \textbf{class} is a [[subset]] of a [[Grothendieck universe]] $U$, while a ([[small category|small]]) \textbf{set} is merely an element of $U$. This gives a relative notion, depending on $U$. As stated here, we get a concept of class like that of the strong theory $MK$; to be more like $BNG$ (and therefore conservative over set theory without an axiom of universes) we should define a \textbf{class} to be a subset of $U$ that is definable in the language of set theory.

\hypertarget{usage}{}\subsection*{{Usage}}\label{usage}

A \textbf{[[large category]]} is a [[category]] whose class of [[morphisms]] is a proper class. It is sufficient that the class of [[objects]] be a proper class, which is also necessary if the category is [[locally small category|locally small]].

Category theorists care about proper classes because many examples of categories in practice (such as [[Set]], to begin with!) are large.

\hypertarget{category_of_classes}{}\subsection*{{Category of classes}}\label{category_of_classes}

The [[category]] of classes $Class$ is a [[large category]] that is not [[locally small]]. It admits all [[colimits]], understood in the following sense.

First, given a class $I$, we define an $I$-indexed family of classes as a map of classes $f:T\to I$. The preimage $f^{-1}(\{i\})$ is precisely the class with index $i$. Such families can be pulled back along maps of classes $J\to I$ and pushed forward along maps $I\to J$.

Next, given a category $I$ (not necessarily locally small), we define an $I$-indexed diagram of classes as a pair $(o:T_O\to Ob(I),m:T_M\to Mor(I))$ of indexed families of classes that satisfies the obvious reformulation of conditions in the definition of a [[functor]].

We now claim that an arbitrary $I$-indexed diagram of classes admits a [[colimit]].

First, the standard reduction of $I$-indexed [[colimits]] to a [[coequalizer]] of a pair of arrows between coproducts indexed by $Mor(I)$ and $Ob(I)$ still works in this context since class-indexed families of classes can be pulled back along source and target maps $Mor(I)\to Ob(I)$.

Secondly, class-indexed coproducts of classes can be computed simply by taking the total class of the corresponding class-indexed family of classes.

Thirdly, [[coequalizers]] of classes exist by [[Scott's trick]]. Observe that given a pair of arrows $f,g:X\to Y$ between classes, we can define an [[equivalence relation]] on $Y$ by saying that $y~y'$ if there is a map $h:[0,n]\to Y$ such that $h(0)=y$, $h(n)=y'$ and for any $i\in[0,n)$ there is $x\in X$ such that $h(i)=f(x)$ and $h(i+1)=g(x)$ or $h(i)=g(x)$ and $h(i+1)=f(x)$. The quotient of $Y$ by this equivalence relation exists by [[Scott's trick]] and is precisely the desired coequalizer.

If one is working in the category of (definable) classes for [[ZF]] or [[ZFC]], or the category of classes of [[NBG]], then all finite \emph{external} diagrams $D\to Class$ have colimits. The reduction to a coequaliser of a pair of finite coproducts works as per usual. \emph{Finite} coproducts exist as one can use finite disjunctions of the defining formulas (in ZF(C)) or Class Separation (in NBG) to define a new class. Coequalisers then exist by the above argument, as [[Scott's trick]] is available due to the class $V$ of sets having well-founded stratifications by sets (for instance the von Neumann [[cumulative hierarchy]] $V = \bigcup_{\alpha\in ORD} V_\alpha$).

\hypertarget{references}{}\subsection*{{References}}\label{references}

A paper detailing one approach to the technical side of how classes appear in [[category theory]] (namely using [[Grothendieck universes]]) is

\begin{itemize}%
\item Paul Blain Levy, \emph{Formulating Categorical Concepts using Classes} (arXiv:\href{https://arxiv.org/abs/1801.08528}{1801.08528})

\end{itemize}
[[!redirects class]] [[!redirects classes]] [[!redirects proper class]] [[!redirects proper classes]]



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