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\section*{semi-left-exact reflection}

\hypertarget{semileftexact_reflections}{}\section*{{Semi-left-exact reflections}}\label{semileftexact_reflections}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{semi-left-exact reflection} (also called a \emph{locally cartesian reflection}) is a [[reflector]] into a [[reflective subcategory]] that preserves some [[pullbacks]]. It fits into a hierarchy of left-exactness properties for reflectors:

$\backslash$begin\{center\} [[left exact reflection|left exact]] $\Rightarrow$ [[reflection with stable units|stable units]] $\Rightarrow$ \textbf{semi-left-exact} $\Rightarrow$ [[simple reflection|simple]] $\backslash$end\{center\}

In particular, semi-left-exactness is sufficient to imply that the corresponding [[reflective factorization system]] exists and can be constructed in one step, and that the reflective subcategory inherits [[local cartesian closure]] from the ambient category.

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

Let $C$ be a [[category]] with [[finite limits]], and $B\subseteq C$ a [[full subcategory|full]] [[reflective subcategory]] with reflector $L:C\to B$. Let $E$ be the class of morphisms inverted by $L$, and let $M$ be the class of morphisms right orthogonal to $E$, i.e. $(E,M)$ is the associated [[reflective prefactorization system]].

The following is a combination of Theorems 4.1 and 4.3 from \hyperlink{CHK}{CHK} with Proposition 1.3 from \hyperlink{GK}{GK}.

$\backslash$begin\{theorem\} The following are equivalent:

\begin{enumerate}%
\item Every pullback of an $E$-morphism along an $M$-morphism is an $E$-morphism.
\item Every pullback of an $E$-morphism along a morphism in $B$ is an $E$-morphism.
\item Every pullback of a reflection unit $\eta_x : x \to L x$ along a morphism in $B$ is an $E$-morphism.
\item $L$ preserves pullbacks along $M$-morphisms.
\item $L$ preserves pullbacks along any morphism in $B$.
\item If $f:x\to y$ is a morphism in $B$, then $f^* : C/y \to C/x$ preserves $E$-morphisms.
\item (If $C$ is locally cartesian closed) If $f:x\to y$ is a morphism in $B$, then $f_* : C/x \to C/y$ maps $B/x$ into $B/y$.

\end{enumerate}
They all imply that $(E,M)$ is a factorization system and that the factorization of an arbitrary morphism $f:x\to y$ can be constructed as the following pullback:

$\backslash$begin\{center\} $\backslash$begin\{tikzcd\} x $\backslash$ardr $\backslash$ardrr,``\{$\backslash$eta\_x\}'' $\backslash$arddr,``f''' $\backslash$ardr,``\{$\backslash$lambda\_f\}'' description $\backslash$ \& $\backslash$bullet $\backslash$arr,``g''' $\backslash$ard,``\{$\backslash$rho\_f\}'' \& L x $\backslash$ard,``L f'' $\backslash$ \& y $\backslash$arr,``\{$\backslash$eta\_y\}''' \& L y $\backslash$end\{tikzcd\} $\backslash$end\{center\} If these conditions hold, we say that the reflector is \textbf{semi-left-exact}. $\backslash$end\{theorem\} $\backslash$begin\{proof\} Since all morphisms in $B$ are $M$-morphisms, (1)$\Rightarrow$(2) and (4)$\Rightarrow$(5), while (2)$\Rightarrow$(3) since the reflection units are in $E$. And since $E$ consists of the $L$-inverted morphisms, (5)$\Rightarrow$(2) and (4)$\Rightarrow$(1). And clearly (6)$\Rightarrow$(2); while for $f\in B$ the action of $f^*$ on a morphism in $C/y$ is the latter's pullback along a pullback of $f$, and any pullback of $f$ lies in $M$; thus (1)$\Rightarrow$(6). So to complete the equivalence of (1)-(6) it will suffice to show that (3) implies (4).

First we prove that (3) implies the factorization exists. For in the diagram above $g$ is a pullback of the unit $y\to L y$ along the morphism $L x \to L y$ in $B$, hence $g\in E$; so by 2-out-of-3 $\lambda_f\in E$, while $\rho_f\in M$ since $M$ is closed under pullback and $L f \in M$.

Now this construction implies that if $f\in M$ then $\lambda_f$ is invertible, hence the naturality square for $\eta$ at $f$ is a pullback. Now if we want to pull back some $h:c\to k$ along an $M$-morphism $f:a\to k$, we can paste two pullback squares to obtain a large pullback rectangle:

$\backslash$begin\{center\} $\backslash$begin\{tikzcd\} d $\backslash$arr $\backslash$ard \& a $\backslash$ard,``f'' $\backslash$arr,``\{$\backslash$eta\_a\}'' \& L a $\backslash$ard,``\{L f\}''$\backslash$ c $\backslash$arr,``h''' \& k $\backslash$arr,``\{$\backslash$eta\_k\}''' \& L k. $\backslash$end\{tikzcd\} $\backslash$end\{center\}

Since $\eta_k \circ h = L h \circ \eta_c$ by naturality, we can now factor this pullback rectangle through the pullback of $f$ along $L h$:

$\backslash$begin\{center\} $\backslash$begin\{tikzcd\} d $\backslash$arr,dashed $\backslash$ard \& e $\backslash$arr $\backslash$ard \& L a $\backslash$ard,``\{L f\}''$\backslash$ c $\backslash$arr,``\{$\backslash$eta\_c\}''' \& L c $\backslash$arr,``\{L h\}''' \& L k $\backslash$end\{tikzcd\} $\backslash$end\{center\}

so that the left-hand square is also a pullback. But this is a pullback of the reflection unit $\eta_c$ along the map $e\to L c$ that lies in $B$ (since $B$ is closed under pullbacks). Thus by (3) the map $d \to e$ lies in $E$, and hence exhibits $e$ as $L d$. The right-hand pullback square above is then $L$ applied to the given pullback square, so that $L$ indeed preserves this pullback, i.e. (4) holds.

Finally, if $C$ is locally cartesian closed, then (6) implies (7) by adjointness, while (7) implies (5) in the form $L/x \circ f^* \cong L/y \circ f^*$ by passage to right-adjoint mates. $\backslash$end\{proof\}

Of course, if $L$ preserves all pullbacks (i.e. it is [[left exact reflection|left exact]]), then it is semi-left-exact. The statement about the existence of the reflective factorization system implies that any semi-left-exact reflection is [[simple reflection|simple]].

$\backslash$begin\{remark\} Note that for any $x\in C$, the functor $L/x\colon C/x \to B/L x$ has a right adjoint given by pullback along $\eta_x : x \to L x$. Condition (3) above then says that the counit of this adjunction is an isomorphism, which is to say that this right adjoint is fully faithful. In this form, semi-left-exactness is equivalent to (a particular case of) the notion of \emph{admissible} reflection in [[categorical Galois theory]]. $\backslash$end\{remark\}

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[semi-left-exact left Bousfield localization]]

\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 (\href{http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_38_03%2FS1446788700023624a.pdf&code=5796045be8904c5183c2e95bce65491e}{pdf})


\item [[David Gepner]] and [[Joachim Kock]], \emph{Univalence in locally cartesian closed categories}, \href{https://arxiv.org/abs/1208.1749}{arxiv:1208.1749}



\end{itemize}
[[!redirects semi-left-exact reflection]] [[!redirects semi-left-exact reflections]] [[!redirects semi left exact reflection]] [[!redirects semi left exact reflections]] [[!redirects semi left-exact reflection]] [[!redirects semi left-exact reflections]] [[!redirects semi-left exact reflection]] [[!redirects semi-left exact reflections]]

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[[!redirects semi-left-exact reflective subcategory]] [[!redirects semi-left-exact reflective subcategories]] [[!redirects semi left exact reflective subcategory]] [[!redirects semi left exact reflective subcategories]] [[!redirects semi left-exact reflective subcategory]] [[!redirects semi left-exact reflective subcategories]] [[!redirects semi-left exact reflective subcategory]] [[!redirects semi-left exact reflective subcategories]]

[[!redirects semi-left-exact localization]] [[!redirects semi-left-exact localizations]] [[!redirects semi left exact localization]] [[!redirects semi left exact localizations]] [[!redirects semi left-exact localization]] [[!redirects semi left-exact localizations]] [[!redirects semi-left exact localization]] [[!redirects semi-left exact localizations]]

[[!redirects semi-left-exact localisation]] [[!redirects semi-left-exact localisations]] [[!redirects semi left exact localisation]] [[!redirects semi left exact localisations]] [[!redirects semi left-exact localisation]] [[!redirects semi left-exact localisations]] [[!redirects semi-left exact localisation]] [[!redirects semi-left exact localisations]]

[[!redirects locally cartesian reflection]] [[!redirects locally cartesian reflections]]

[[!redirects locally cartesian reflector]] [[!redirects locally cartesian reflectors]]

[[!redirects locally cartesian localization]] [[!redirects locally cartesian localizations]]

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