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\begin{document}

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\section*{pure subobject}

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

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

[[!include category theory - contents]]

\hypertarget{compact_objects}{}\paragraph*{{Compact objects}}\label{compact_objects}

[[!include compact object - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A pure subobject is a [[monomorphism]] $A \rightarrowtail B$ -- hence a [[subobject]] $A$ of some [[object]] $B$ in some [[category]] -- which is a \emph{[[pure morphism]]}: such that any sufficiently small system of [[equations]] involving constants in $A$ that admits a solution in $B$ also admits a solution in $A$. This generalises the classical notions of `pure group' and `pure submodule'.

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

Let $\kappa$ be a regular [[cardinal]]. A \textbf{$\kappa$-[[pure morphism]]} in a [[category]] $\mathcal{C}$ is a morphism $f : A \to B$ with the following extension property:

\begin{itemize}%
\item Given any morphisms $f' : A' \to B'$, $a : A' \to A$, $b : B' \to B$ in $\mathcal{C}$, if both $A'$ and $B'$ are $\kappa$-[[compact object|compact]] and $f \circ a = b \circ f'$,

\begin{displaymath}
\itexarray{
    A' &\stackrel{a}{\to}& A
    \\
    \downarrow^{\mathrlap{f'}} && \downarrow^{\mathrlap{f}}
    \\
    B' &\stackrel{b}{\to}& B
  }
  \,,
\end{displaymath}
then there exists a (not necessarily unique) morphism $\bar{a} : B' \to A$ in $\mathcal{C}$ such that $a = \bar{a} \circ f'$. (We do not assert any compatibility with $b$, however.)



\end{itemize}
A \textbf{$\kappa$-pure subobject} is a $\kappa$-pure monomorphism.

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

\begin{itemize}%
\item A [[retract]] is a $\kappa$-pure subobject in any category, for any $\kappa$.


\item Conversely, any $\kappa$-pure subobject in [[Set]] is a retract.


\item If $A$ is an [[injective]] [[module]] and $B$ is any module containing $A$ as a submodule, then the inclusion $A \hookrightarrow B$ is $\kappa$-pure. (This can be checked directly without recourse to the fact that any injective submodule is a retract!)


\item The torsion subgroup of any [[abelian group]] is a $\kappa$-pure subgroup, since it is a filtered colimit of [[direct sum|direct summands]]. (See below.)



\end{itemize}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\begin{lemma}
\label{}\hypertarget{}{}
In any category:

\begin{itemize}%
\item The class of $\kappa$-pure morphisms is closed under composition.


\item If $g \circ f$ is a $\kappa$-pure morphism, then so is $f$.


\item If $\kappa' \le \kappa$, then any $\kappa$-pure morphism is also $\kappa'$-pure.



\end{itemize}
\end{lemma}
\begin{prop}
\label{}\hypertarget{}{}
In a $\kappa$-[[accessible category]], any $\kappa$-pure morphism is necessarily monic.

\end{prop}
This is \hyperlink{AdamekRosicky}{LPAC, Prop. 2.29}.

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ is a $\kappa$-accessible category, then $\kappa$-pure subobjects in $\mathcal{C}$ are closed under $\kappa$-[[filtered limit|filtered colimits]] in the [[arrow category]] $Arr (\mathcal{C})$.

If $\mathcal{C}$ is a $\kappa$-accessible category with [[pushout|pushouts]], then any $\kappa$-pure subobject in $\mathcal{C}$ is a $\kappa$-filtered colimit in $Arr (\mathcal{C})$ of retracts in $\mathcal{C}$.

\end{prop}
This is \hyperlink{AdamekRosicky}{LPAC, Prop. 2.30}.

\begin{prop}
\label{}\hypertarget{}{}
In a locally $\kappa$-[[locally presentable category|presentable category]], every $\kappa$-pure morphism is a [[regular monomorphism]].

\end{prop}
This is \hyperlink{AdamekRosicky}{LPAC, Prop. 2.31}.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{theorem}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a $\kappa$-accessible category, and let $\mathcal{D}$ be a full subcategory of $\mathcal{C}$ that is closed under $\kappa$-filtered colimits for some regular cardinal $\kappa$. Then, $\mathcal{D}$ is a $\mu$-accessible category for some regular cardinal $\mu$ sharply larger than $\kappa$ if and only if $\mathcal{D}$ is closed under $\kappa$-pure subobjects in $\mathcal{C}$.

In particular, a category $\mathcal{D}$ is accessible if and only if there is a fully faithful functor $R : \mathcal{D} \to Set^{\mathcal{A}}$ where $\mathcal{A}$ is small, $R$ [[created limit|creates]] colimits for all $\kappa$-filtered diagrams, and $\mathcal{D}$ is closed under $\kappa$-pure subobjects in $Set^{\mathcal{A}}$.

\end{theorem}
This is \hyperlink{AdamekRosicky}{LPAC, Cor. 2.36}.

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

\begin{itemize}%
\item LPAC], [[Jiri Rosicky]], \emph{[[Locally presentable and accessible categories]]}

\end{itemize}
[[!redirects pure morphism]] [[!redirects pure monomorphism]]

[[!redirects pure subobjects]] [[!redirects pure morphisms]] [[!redirects pure monomorphisms]]



\end{document}