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\begin{document}

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\section*{locally bounded category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{context_2}{Context}\dotfill \pageref*{context_2} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{features}{Features}\dotfill \pageref*{features} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Local boundedness of a [[category]] is a generalization of the notion of [[locally presentable category|local presentability]] that includes the category of [[topological spaces]].

\hypertarget{context_2}{}\subsection*{{Context}}\label{context_2}

Let $C$ be a [[small category|small]] [[cocomplete category]] with a proper [[factorization system]], i.e., an [[orthogonal factorization system]] $(E,M)$ where every [[morphism|map]] in $E$ is an [[epimorphism]] and every map in $M$ is a [[monomorphism]].

The $M$-\textbf{union} of a small family of $M$-[[subobjects]] $(A_j \to B)_{j \in J}$ is the unique $M$-subobject $A \to B$ containing the $A_j$ and so that the induced map $\sum_j A_j \to A$ is in $E$. The union is calculated by applying the $(E,M)$ factorization to the canonical map $\sum_j A_j \to B$.

If the map $\sum_j A_j \to B$ is in $E$, we say $(A_j \to B)_{j \in J}$ is an $M$-\textbf{union}. The set $J$ is a preorder under the relation $j \leq k$ if $A_j \leq A_k$ as $M$-subobjects of $B$. Regarding the $A_j$ as a diagram of shape $J$, the family $(A_j \to B)_{j \in J}$ is an $M$-union if and only if the map colim$A_j \to B$ is in $E$. We say $(A_j \to B)_{j \in J}$ is a \textbf{filtered union of} $M$-\textbf{subobjects} if it is a union of $M$-subobjects and if the category $J$ is [[filtered category|filtered]].

A [[representable functor]] $C(X,-)$ \textbf{preserves} the $M$-union of $(A_j \to B)_{j \in J}$ if the functions $C(X,A_j) \to C(X,A)$ are jointly surjective, i.e., if each $X \to A$ factors through some $X \to A_j$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\lambda$ be a [[regular cardinal]], and let $C$ be a cocomplete category with a proper factorization system $(E,M)$.

\begin{defn}
\label{}\hypertarget{}{}
An object $X$ in $C$ is $\lambda$-\textbf{bounded} if $C(X,-)$ preserves $\lambda$-[[filtered colimit|filtered]] [[unions]] of $M$-subobjects.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A small set $G$ of objects of $C$ is an $(E,M)$-\textbf{generator} if $f \colon A \to B$ in $M$ is invertible whenever $f_* \colon C(X,A) \to  C(X,B)$ is [[bijection|bijective]] for all $X \in G$. Equivalently, $G$ is an $(E,M)$-\textbf{generator} if for each $A \in C$ the family of maps $X \to A$ is jointly in $E$, i.e., if the map $\sum_{X \in G} \sum_{C(X,A)} X \to A$ is in $E$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A [[category]] $C$ is called \textbf{locally} $\lambda$-\textbf{bounded} with respect to a proper factorization system $(E,M)$ if

\begin{itemize}%
\item it has an $(E,M)$-generator $G$ each of whose objects is $\lambda$-bounded


\item it has arbitrary cointersections (even large ones) of maps in $E$ --- that is, it is [[E-cocomplete category|E-cocomplete]].



\end{itemize}
\end{defn}
\hypertarget{features}{}\subsection*{{Features}}\label{features}

\begin{prop}
\label{}\hypertarget{}{}
In a locally $\lambda$-presentable category, every $\lambda$-presentable object is $\lambda$-bounded. Hence a $\lambda$-presentable category is $\lambda$-bounded.

\end{prop}
\begin{proof}
This appears as Lemma 2.3.1 of \hyperlink{FreydKelly}{Freyd-Kelly}

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Locally bounded categories are necessarily complete

\end{prop}
\begin{proof}
This appears as Corollary 2.2 of \hyperlink{KellyLack}{Kelly-Lack}.

The essential point is an $(E,M)$-variant of the special adjoint functor theorem: if $C$ is cocomplete, has a proper factorization system $(E,M)$, admits arbitrary $E$-cointersections, and has an $(E,M)$-generator, then every cocontinuous functor $C \to D$ has a right adjoint.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The following examples are discussed in Section 6.1 of Kelly's \emph{Basic concepts of enriched category theory}.

\begin{itemize}%
\item [[locally presentable category|Locally presentable categories]] such as [[simplicial sets]], [[categories]], [[abelian groups]], [[Set|sets]].


\item [[compactly generated topological space|Compactly generated spaces]], and likewise based compactly generated spaces, with $E$ the surjections and $M$ the subspace inclusions. The point is an $(E,M)$-generator.


\item [[quasi-topological space|Quasi-topological spaces]]. Note that this category is not $E$-well-copowered.


\item [[Banach spaces]] with $E$ the epimorphisms, equivalently the dense maps, and $M$ the [[extremal monomorphisms]], equivalently the inclusions of closed subspaces with the induced norm. The base field is an $(E,M)$-generator.



\end{itemize}
\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

The term \textbf{locally ranked} is sometimes used to refer to a locally bounded category which in addition is co-wellpowered. For example, this terminology is used in \hyperlink{AHRT}{Ad\'a{}mek et. al.}.

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

The contents of this page are taken from:

\begin{itemize}%
\item [[Max Kelly]], [[Steve Lack]], \emph{$V$-cat is locally presentable or locally bounded if $V$ is so} TAC (2001)

\end{itemize}
See also:

\begin{itemize}%
\item [[Max Kelly]], \emph{Basic concepts of enriched category theory}.


\item [[Peter Freyd]], [[Max Kelly]], \emph{Categories of continuous functors} J. Pure. Appl. Algebra 2 (1972) 169-191.



\end{itemize}
\begin{itemize}%
\item [[Jírí Adámek]], [[Horst Herrlich]], [[Jírí Rosickỳ]], [[Walter Tholen]], \emph{On a generalized small-object argument for the injective subcategory problem}. Cah. Topol. G\'e{}om. Diff\'e{}r. Cat\'e{}g 43 (2002) 83--106.

\end{itemize}
[[!redirects locally bounded categories]]



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