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\section*{semi-left-exact left Bousfield localization}

\hypertarget{semileftexact_left_bousfield_localizations}{}\section*{{Semi-left-exact left Bousfield localizations}}\label{semileftexact_left_bousfield_localizations}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

When constructing a general [[left Bousfield localization]] of a [[cofibrantly generated model category]] $C$ at a set of maps $S$, it is fairly straightforward to construct a [[localization]] functor, i.e. a [[fibrant replacement]] functor for the putative localized [[model category]] $L_S C$, by a [[small object argument]]. We represent the maps in $S$ by cofibrations between [[cofibrant]] objects, take their [[pushout]] products with all boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ (assuming for simplicity that $C$ is a [[simplicial model category]]), and add all the generating [[acyclic cofibrations]] of $C$. Then an object $Z$ has the right lifting property with respect to this set if and only if it is [[fibrant]] in $C$ (because we added the generating acyclic cofibrations of $C$) and for all $f:A\to B$ in $S$ the induced map $Map(B,Z) \to Map(A,Z)$ of simplicial mapping spaces (which is a fibration since $f$ is a cofibration and $Z$ is fibrant) is an acyclic fibration, i.e. $Z$ is $S$-local. However, it is much harder to construct a factorization of an \emph{arbitrary} map as an $S$-acyclic cofibration followed by an $S$-fibration --- one has to use a cardinality argument to obtain a set of generating $S$-acyclic cofibrations --- and accordingly the $S$-fibrations have no explicit description.

This can be seen as a homotopy-theoretic analogue of the construction of a [[reflective factorization system]]. The localization functor exhibits the $S$-local objects as a [[reflective subcategory]], while the $S$-acyclic cofibrations and $S$-fibrations are a model-categorical representation of the corresponding reflective [[factorization system]], whose left class consists of the morphisms inverted by the [[reflector]] (here, the $S$-local equivalences) and whose right class is defined by orthogonality to these. Even in [[1-category theory]], constructing reflective factorizations requires finicky cardinality or size-based arguments as well. However, there are some reflections, such as the [[semi-left-exact reflections]], for which the [[reflective factorization system]] can be constructed by a direct argument in one step. The homotopy-theoretic analogue of these is a \textbf{semi-left-exact left Bousfield localization}.

\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

The following is Theorem 1.1 of \hyperlink{Stanculescu}{Stanculescu}, which is an improved version of Theorems 9.3 and 9.7 from \hyperlink{Bousfield01}{Bousfield 2001}, which are in turn an improved version of Appendix A of \hyperlink{BousfieldFriedlander78}{Bousfield-Friedlander 78}.

$\backslash$begin\{theorem\} Let $C$ be a [[model category]] and $Q:C\to C$ a functor equipped with a natural transformation $\alpha:Id\to Q$ such that

\begin{enumerate}%
\item $Q$ is a [[homotopical functor]], i.e. preserves weak equivalences.
\item For each $X\in C$, the map $Q \alpha_X:Q X \to Q Q X$ is a weak equivalence, and the map $\alpha_{Q X}:Q X \to Q Q X$ becomes a [[monomorphism]] in the [[homotopy category]].
\item Define an object $X$ to be $Q$-local if it is fibrant and $X\to Q X$ is a weak equivalence, and define a morphism $f$ to be a $Q$-equivalence if $Q f$ is a weak equivalence. Then pullback along fibrations between $Q$-local objects preserves $Q$-equivalences.

\end{enumerate}
Then there is a new model structure $C^Q$ on $C$ whose cofibrations are those of $C$, whose weak equivalences are the $Q$-equivalences, and whose fibrations are the maps whose $\alpha$-naturality-square is a [[homotopy pullback]]. Moreover, $C^Q$ is [[proper model category|right proper]], and simplicial if $C$ is. $\backslash$end\{theorem\}

The first two conditions say essentially that $Q$ is a homotopical reflection into some subcategory (namely the $Q$-local objects). The third condition says that it is semi-left-exact (a 1-categorical reflection of $C$ into $B\subseteq C$ is semi-left-exact if and only if pullback along morphisms in $B$ preserves morphisms that are inverted by the reflector).

For details of the proof, see the references. The central point is the construction of the factorization into an $S$-acyclic cofibration and an $S$-fibration, which proceeds by first applying $Q$ along with fibrant replacement, then taking a homotopy pullback: the same way that a [[reflective factorization system]] is constructed from a [[semi-left-exact reflection]] in 1-category theory.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item Right properness of semi-left-exact left Bousfield localizations was also shown in \hyperlink{GK}{Gepner-Kock, Prop. 7.8}, with special attention paid to [[type-theoretic model categories]].


\item Nullification, i.e. localization at a family of maps $A\to \ast$, is always semi-left-exact. (Indeed, nullification is what in [[homotopy type theory]] is called a [[higher modality]], and has [[stable units|reflection with stable units]], a stronger condition.)


\item [[left-exact reflection|Left exact]] reflections are always semi-left exact. In particular, the left-exact localizations that present [[Grothendieck (∞,1)-toposes]] can be constructed in this way.



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

\begin{itemize}%
\item [[Aldridge Bousfield]], [[Eric Friedlander]], def. 1.1.6 in \emph{Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets}, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf}{pdf})


\item [[Aldridge Bousfield]], \emph{On the telescopic homotopy theory of spaces}, Trans. Amer. Math. Soc. 353 (2001), 2391-2426, \href{https://www.ams.org/journals/tran/2001-353-06/S0002-9947-00-02649-0/}{web with fulltext}


\item [[Alexandru E. Stanculescu]], \emph{Note on a theorem of Bousfield and Friedlander}, Topology and its Applications 155(13), \href{https://arxiv.org/abs/0806.4547}{arxiv:0806.4547}.


\item [[Philip Hirschhorn]], chapter 13 of \emph{Model Categories and Their Localizations}, 2003 (\href{http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99}{AMS}, \href{http://www.gbv.de/dms/goettingen/360115845.pdf}{pdf toc}, \href{http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf}{pdf})


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

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



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