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\begin{document}

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\section*{orthogonal subcategory problem}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{theorem_in_the_setting_of_locally_presentable_categories}{Theorem in the setting of locally presentable categories}\dotfill \pageref*{theorem_in_the_setting_of_locally_presentable_categories} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{orthogonal subcategory problem} for a [[class]] of [[morphism]]s $\Sigma$ in a [[category]] $C$ asks whether the [[full subcategory]] $\Sigma^\perp$ of [[object]]s [[orthogonal]] to $\Sigma$ is a [[reflective subcategory]].

Here, an object $X$ is said to be orthogonal to a map $f:A\to B$ if every morphism $A\to X$ factors uniquely through $f$. For example, in a category of presheaves, $X:C^{op}\to Set$ is orthogonal to the canonical map $colim_i C(-,a_i)\to C(-,\colim_i a_i)$ if it preserves the limit $\lim_i a_i$ in $C^{op}$.

This problem is related to the problem of [[localization]]. Suppose $\Sigma^\perp$ is indeed a [[reflective subcategory]]; let $r: C \to \Sigma^\perp$ be the reflector (the [[left adjoint]] to the inclusion $i: \Sigma^\perp \to C$).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{ulemma}
If $f: a \to b$ belongs to $\Sigma$, then $r(f): r(a) \to r(b)$ is an [[isomorphism]] in $\Sigma^\perp$.

\end{ulemma}
\begin{proof}
By definition of [[orthogonality]], for every object $X$ in $\Sigma^\perp$, $f$ induces an isomorphism of [[hom-set]]s

\begin{displaymath}
\hom_C(f, i(X)): \hom_C(B, i(X)) \stackrel{\sim}{\to} \hom_C(A, i(X))
\end{displaymath}
Since $r \dashv i$, this means that for all $X$ in $\Sigma^\perp$ the map

\begin{displaymath}
\hom_{\Sigma^\perp}(r(f), X): \hom_{\Sigma^\perp}(r(B), X) \to \hom_{\Sigma^\perp}(r(A), X)
\end{displaymath}
is an [[isomorphism]], so that $\hom_{\Sigma^\perp}(r(f), -)$ is a [[natural isomorphism]] between [[representable functor|representables]]. By the [[Yoneda lemma]], this means $r(f)$ is an isomorphism.

\end{proof}
This lemma can be sharpened. First, given a category $C$, there is a [[Galois connection]] between classes of morphisms $\Sigma$ and classes of objects $K$, induced by the predicate that $\hom_C(\sigma, k)$ is a bijection for all $\sigma \in \Sigma$ and all $k \in K$. The induced monad on the collection of morphism classes $\Sigma$ (partially ordered by inclusion) may be called ``saturation'', denoted $sat(\Sigma)$. An easy argument due to Gabriel-Zisman is that if $C$ is finitely [[cocomplete category|cocomplete]], then $(C, sat(\Sigma))$ satisfies the axioms for a [[calculus of fractions]], so that the [[localization]] $C[(sat(\Sigma))^{-1}]$ can be constructed. An easy argument then establishes the following sharpened lemma:

\begin{ulemma}
If $\Sigma^\perp \hookrightarrow C$ is [[reflective subcategory|reflective]], then the reflection $r: C \to \Sigma^\perp$ exhibits a canonical [[equivalence of categories|equivalence]]

\begin{displaymath}
C[(sat(\Sigma))^{-1}] \simeq \Sigma^\perp
\end{displaymath}
\end{ulemma}
\begin{ulemma}
If $\Sigma^\perp \hookrightarrow C$ is [[reflective subcategory|reflective]], then the reflection $r: C \to \Sigma^\perp$ exhibits a canonical [[equivalence of categories|equivalence]]

\begin{displaymath}
C[(sat(\Sigma))^{-1}] \simeq \Sigma^\perp
\end{displaymath}
\end{ulemma}
To be connected with things like [[small object argument]], [[Bousfield localization]], and others\ldots{}

\hypertarget{theorem_in_the_setting_of_locally_presentable_categories}{}\subsection*{{Theorem in the setting of locally presentable categories}}\label{theorem_in_the_setting_of_locally_presentable_categories}

\begin{theorem}
\label{}\hypertarget{}{}
If $\Sigma$ is a set of morphisms in a [[locally presentable]] category $\mathcal{A}$ then the orthogonality subcategory $\Sigma^\top$ is reflective in $\mathcal{A}$.

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

\begin{itemize}%
\item [[Gabriel-Zisman]]


\item [[Peter Freyd]], [[Max Kelly]], \emph{Categories of continuous functors I}, Jour. Pure Appl. Alg. 2 (1972), 169-191. (\href{http://www.sciencedirect.com/science/article/pii/0022404972900011}{web})


\item [[Jiří Adámek]], [[Jiří Rosický]], \emph{[[Locally presentable and accessible categories]]}, Cambridge University Press, (1994). But see the \href{https://www.tu-braunschweig.de/Medien-DB/iti/cor.pdf}{errata}.



\end{itemize}


\end{document}