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\begin{document}

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\section*{M-complete category}

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

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

[[!include category theory - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{complete_and_cocomplete_categories}{}\section*{{$M$-complete and $E$-cocomplete categories}}\label{complete_and_cocomplete_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{construction_of_factorization_systems}{Construction of factorization systems}\dotfill \pageref*{construction_of_factorization_systems} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[category]] is $M$-complete or $E$-cocomplete if has certain [[limits]] or [[colimits]] of [[morphisms]] in a given class $M$ or $E$.

Not to be confused with an [[M-category]].

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

Let $C$ be a [[category]] and let $M$ be a class of [[monomorphisms]] in $C$. (Often, $M$ will be the right class in an [[orthogonal factorization system]].) We say that $C$ is \textbf{$M$-complete} if it admits all (even [[large category|large]]) [[intersections]] of $M$-[[subobjects]]. This means that it admits all (even large) [[wide pullbacks]] of families of $M$-morphisms, and such pullbacks are again in $M$. (If $M$ is the right class of an OFS, then any intersection of $M$-morphisms which exists is automatically in $M$.)

If $M$ is the class of all monomorphisms, we may say \textbf{mono-complete} for $M$-complete.

Dually, if $E$ is a class of [[epimorphisms]], we say $C$ is \textbf{$E$-cocomplete} if it admits all [[cointersection]]s of $E$-morphisms, and \textbf{epi-cocomplete} if $E$ is the class of all epimorphisms.

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

\begin{itemize}%
\item If $C$ is $M$-[[well-powered category|well-powered]], then no large limits are required in the definition of $M$-completeness. Therefore, if $C$ is well-powered and [[complete category|complete]], it is $M$-complete whenever $M$ is the right class in an OFS. Dually, if $C$ is well-copowered and cocomplete, it is $E$-cocomplete whenever $E$ is the left class in an OFS.


\item For similar reasons, the category [[FinSet]] is mono-complete and epi-cocomplete---although it is not complete or cocomplete, it is [[finitely complete category|finitely complete]] and [[finitely cocomplete category|cocomplete]], and its subobject lattices and quotient lattices are likewise [[essentially finite category|essentially finite]].


\item If $C$ is a [[topological concrete category]] over a category $D$ which is mono-complete or epi-complete, then $C$ is also mono-complete or epi-complete. For the faithful forgetful functor $U\colon C\to D$ preserves and reflects monos and epis, and so the [[weak structure|initial]] $C$-structure on an [[intersection]] of underlying monos in $D$ gives an intersection in $C$ and the [[strong structure|final]] $C$-structure on a [[cointersection]] of underlying epis in $D$ gives a cointersection in $C$.



\end{itemize}
\hypertarget{construction_of_factorization_systems}{}\subsection*{{Construction of factorization systems}}\label{construction_of_factorization_systems}

$M$-completeness is useful for constructing [[orthogonal factorization systems]]. The following is Lemma 3.1 in \hyperlink{CHK}{CHK}.

\begin{theorem}
\label{ConstructingOFS}\hypertarget{ConstructingOFS}{}
Let $M$ be a class of maps in a category $C$, and assume that

\begin{enumerate}%
\item $M$ consists of monomorphisms,
\item $M$ is closed under [[composition]],
\item all [[pullbacks]] of $M$-morphisms exist in $C$ and are again in $M$, and
\item $C$ is $M$-complete.

\end{enumerate}
Then there is an orthogonal factorization system $(E,M)$, with $E = {}^\perp M$.

\end{theorem}
\begin{proof}
Given $f\colon A\to B$, let $m$ be the intersection of all $M$-morphisms $n\colon X \to B$ through which $f$ factors. Then by the universal property of this intersection, we have $f = m e$ for some $e$; thus it suffices to show $e\in E$. Suppose given a commutative square

\begin{displaymath}
\itexarray{ A & \overset{g}{\to} & Z \\
  ^e \downarrow & & \downarrow^p \\
  Y & \underset{h}{\to} & W}
\end{displaymath}
with $p\in M$. By pulling $p$ back to $Y$ (since pullbacks of $M$-morphisms exist), we may assume that $Y=W$ and $h$ is the identity. But now the composite $m p$ is an $M$-morphism through which $f$ factors, so by definition, $m$ factors through it. Thus $p$ is an isomorphism and so the lifting problem can be solved.

\end{proof}
In fact, it is easy to see that the same proof constructs a [[factorization structure for sinks]].

Note that if $M$ is already part of a [[prefactorization system]], then any composite, pullback, or intersection of $M$-morphisms which exists is automatically also in $M$, since $M = E^\perp$.

\begin{cor}
\label{}\hypertarget{}{}
Let $(E,M)$ be a [[prefactorization system]] on a category $C$, and assume that

\begin{enumerate}%
\item $M$ consists of monomorphisms,
\item All pullbacks of $M$-morphisms exist in $C$, and
\item $C$ is $M$-complete.

\end{enumerate}
Then $(E,M)$ is an orthogonal factorization system.

\end{cor}
The following is a slight generalization of Theorem 3.3 of \hyperlink{CHK}{CHK}. There it is stated only for the case $M=$ [[strong monomorphisms]], in which case a finitely complete and $M$-complete category is called \textbf{finitely well-complete}.

\begin{theorem}
\label{OFSFromAdjunction}\hypertarget{OFSFromAdjunction}{}
Let $S : A \rightleftarrows C : T$ be an [[adjunction]], and assume that $A$ is finitely complete and $M$-complete for some OFS $(E,M)$, where $M$ consists of monomorphisms and contains the split monics. Define $E_S$ to be the class of maps inverted by $S$, and $M_S = (E_S)^\perp$; then $(E_S,M_S)$ is an OFS on $A$.

\end{theorem}
\begin{proof}
First of all, since $M_S$ belongs to a prefactorization system, it is closed under composites, pullbacks, and any intersections which exist. Therefore, if we define $M' \coloneqq M \cap M_S$, then $M'$ satisfies the hypotheses of Theorem \ref{ConstructingOFS}, and so we have an OFS $(E',M')$.

Moreover, it is useful to notice that $E_S={}^\perp T(\hom(C))$: this is an easy consequence of the fact that if $S\dashv T$, then $S a\perp b\iff a\perp T b$, since $f\perp T u\iff S f\perp u$ for each $u\in\hom(C)$, so that $S f$ is an isomorphism.

Now suppose given $f\colon A\to B$; we want to construct an $(E_S,M_S)$-factorization. Let $v$ be the pullback of $T S f$ along the [[unit of an adjunction|unit]] $\eta_B \colon B \to T S B$. The naturality square for $\eta$ at $f$ shows that $f$ factors through $v$, say $f = v w$.

\begin{displaymath}
\begin{array}{ccccc}
A & \\
 &\overset{w}\searrow\\
 && P &\overset{u}\to& T S A \\
 && {}^v\downarrow  && \downarrow^{T S f}\\
 && B &\underset{\eta_B}\to& T S B
\end{array}
\end{displaymath}
Since $T S f$ is evidently in $M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C))$, so is $v$; thus it suffices to find an $(E_S,M_S)$-factorization of $w$.

Let $w = n g$ be the $(E',M')$-factorization of $w$. Since $M' \subseteq M_S$, it suffices to show that $g\in E_S$. Note also that since $w$ is a first factor of the unit $\eta_A$, by passing to adjuncts we find that $S w$ is [[split monic]]: in the former diagram we have $u w=\eta_A$, so that the adjunct $\epsilon_{S A} \cdot S u\cdot  S n\cdot  S g=1$, hence also $S g$ is a split monic. But $T S g$ is then also split monic, hence belongs to $M$ and thus also to $M'$ (since it obviously belong to $M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C))$). Therefore, since $g\in E'$, the naturality square for $\eta$ at $g$ contains a lift: there is an $\alpha\colon X\to T S A$ such that in the diagram

\begin{displaymath}
\begin{array}{ccc}
A & \overset{\eta_A}\to & T S A \\
{}^g\downarrow && \downarrow^{T S g} \\
X &\overset{\eta_X}\to& T S X 
\end{array}
\end{displaymath}
$\alpha\cdot g=\eta_A$ and $T S g \cdot \alpha=\eta_X$. Passing to adjuncts again, we find that $S g$ is also [[split epic]], since we can consider the diagram

\begin{displaymath}
\begin{array}{ccccc}
S A & \overset{S\eta_A}\longrightarrow & S T S A &\overset{\epsilon_{S A}}\longrightarrow & S A\\
{}^{S g}\downarrow && \phantom{aaa}\downarrow_{S T S g} &&\downarrow^{S g}\\
S X &\underset{S\eta_X}\longrightarrow & S T S X &\underset{\epsilon_{S X}}\longrightarrow & S X
\end{array}
\end{displaymath}
and the commutativity

\begin{displaymath}
S g \cdot \epsilon_{S A} \cdot S\alpha = \epsilon_{S X} \cdot  S T S g \cdot S\alpha = \epsilon_{S X}\cdot S\eta_X = 1
\end{displaymath}
Hence $S g$ is an isomorphism; thus $g\in E_S$ as desired.

\end{proof}
This is useful in the construction of [[reflective factorization systems]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[complete category]]


\item [[locally bounded category]]



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

\begin{itemize}%
\item Cassidy and H\'e{}bert and [[Max Kelly|Kelly]], ``Reflective subcategories, localizations, and factorization systems''. \emph{J. Austral. Math Soc. (Series A)} 38 (1985), 287--329

\end{itemize}
[[!redirects M-complete category]] [[!redirects M-complete categories]]

[[!redirects mono-complete category]] [[!redirects mono-complete categories]]

[[!redirects E-cocomplete category]] [[!redirects E-cocomplete categories]]

[[!redirects epi-cocomplete category]] [[!redirects epi-cocomplete categories]]

[[!redirects finitely well-complete category]] [[!redirects finitely well-complete categories]]



\end{document}