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\begin{document}

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\section*{quotient object}

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{galois_connection_between_quotients_and_relations}{Galois connection between quotients and relations}\dotfill \pageref*{galois_connection_between_quotients_and_relations} \linebreak
\noindent\hyperlink{InToposes}{In toposes}\dotfill \pageref*{InToposes} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{quotient object} $Q$ of a [[congruence]] (an [[internalization|internal]] [[equivalence relation]]) $E$ on an [[object]] $X$ in a [[category]] $C$ is the [[coequalizer]] $Q$ of the induced pair of morphisms $E \underoverset{\longrightarrow}{\longrightarrow}{} X$.

If $E$ is additionally the [[kernel pair]] of the map $X \to Q$, then $Q$ is called an \textbf{effective quotient} -- and $E$ is called an \textbf{effective} congruence, with the map $X \to Q$ being an [[effective epimorphism]] (terminology as for \emph{[[effective group action]]}).

Sometimes the term is used more loosely to mean an arbitrary [[coequalizer]]. It may also refer to a co-[[subobject]] of $X$ (that is, a subobject of $X$ in the [[opposite category]] $C^\op$), without reference to any congruence on $X$. Note that in a [[regular category]], any [[regular epimorphism]] (i.e. a ``regular quotient'' in the co-subobject sense) is in fact the quotient (= coequalizer) of its [[kernel pair]] (actually, we can prove this under weaker hypotheses; see below).

\hypertarget{galois_connection_between_quotients_and_relations}{}\subsection*{{Galois connection between quotients and relations}}\label{galois_connection_between_quotients_and_relations}

As we have said, there are various notions of quotient object. Let us consider the most general one, so that $Quot(X)$ of an object $X$ denotes the poset of co-subobjects of $X$, in other words the [[poset|posetal]] [[reflection]] of the [[preorder]] of epis $X \to Q$ which is a [[full subcategory]] of the [[co-slice category]] $X \downarrow \mathbf{C}$. A \emph{regular quotient} then refers to a regular epi $X \to Q$.

On the other hand, let $Rel(X)$ be the poset of [[relations]] on $X$, i.e., the poset of subobjects of $X \times X$, or in other words the posetal reflection of the preorder of monos $i = \langle e_1, e_2 \rangle: E \rightarrowtail X \times X$ which is a full subcategory of the [[slice category]] $\mathbf{C} \downarrow X \times X$.

Between $Quot(X)$ and $Rel(X)$ there is a relation $\perp$ where $q \perp \langle e_1, e_2 \rangle$ means exactly $q \circ e_1 = q \circ e_2$. If the coequalizer $coeq(e_1, e_2)$ of the parallel pair $e_1, e_2: E \rightrightarrows X$ exists, then by definition we have $coeq(e_1, e_2) \leq q$ iff $q e_1 = q e_2$. On the other hand, if the [[kernel pair]] $\ker(q)$ of $q$ exists, then by definition we have $q e_1 = q e_2$ iff $\langle e_1, e_2 \rangle \leq \ker(q)$.

This indicates in a category which admits coequalizers and kernel pairs, we have

\begin{prop}
\label{}\hypertarget{}{}
$coeq: Rel(X) \to Quot(X)$ is [[left adjoint]] to $\ker: Quot(X) \to Rel(X)$.

\end{prop}
Or, in other words, that $\ker$ and $coeq$ set up a [[Galois connection]] between $Quot(X)^{op}$ and $Rel(X)$.

Restricting consideration to kernel pairs only of epis, or coequalizers only of jointly monic pairs, is no real restriction in the presence of epi-mono factorizations:

\begin{lemma}
\label{image}\hypertarget{image}{}
In a category where every morphism $f: A \to B$ has an [[image|epi-mono factorization]] $f = i \circ q$, we have $\ker(f) = \ker(q)$. Similarly, for a pair $f, g: X \rightrightarrows Y$, we have $coeq(f, g) = coeq(e_1, e_2)$ where $\langle f, g \rangle: X \to Y \times Y$ factors as an epi $p: X \to E$ followed by a mono $\langle e_1, e_2 \rangle: E \to Y \times Y$.

\end{lemma}
\begin{proof}
We prove just the first statement; the second is proven similarly. It suffices to observe that the same class of jointly monic pairs $(e_1, e_2)$ are coequalized by $f$ as by $q$; the kernel pair is by definition the maximum of this class. If $q e_1 = q e_2$, then by applying $i$ to both sides we deduce $f e_1 = f e_2$. If $f e_1 = f e_2$, i.e., if $i q e_1 = i q e_2$, then $q e_1 = q e_2$ by monicity of $i$.

\end{proof}
\begin{prop}
\label{galois}\hypertarget{galois}{}
Suppose $\mathbf{C}$ is a category with coequalizers and kernel pairs and where every morphism has an epi-mono factorization. Then every regular epi $q$ is the coequalizer of its kernel pair: $q = coeq \circ \ker(q)$. And every kernel pair is the kernel pair of its coequalizer: $i = \ker \circ coeq(i)$.

\end{prop}
\begin{proof}
We just prove the first statement; the second is proved similarly. We have of course a [[counit]] $coeq \circ \ker(q) \leq q$. On the other hand, if $q = coeq(f, g)$ (where we may assume $\langle f, g \rangle$ is monic by the lemma), then we have a unit $\langle f, g \rangle \leq \ker \circ coeq(f, g) = \ker(q)$; applying $coeq$ to each side, we have $q \leq coeq \circ \ker(q)$, as desired.

\end{proof}
\hypertarget{InToposes}{}\subsection*{{In toposes}}\label{InToposes}

Constructing quotient objects in an [[elementary topos]] $\mathbf{E}$, starting from one or another standard definition (e.g., [[finitely complete category]] with [[power objects]]) that doesn't already mention colimits, is not trivial.

The standard approach seen in textbooks (see for example \emph{[[Sheaves in Geometry and Logic]]}), apparently first introduced by C.J. Mikkelsen but first published by \hyperlink{RP}{Par\'e{}}, is to prove that the contravariant [[power object]] functor $P \colon \mathbf{E}^{op} \to \mathbf{E}$ is [[monadic functor|monadic]]. It follows that $P$ [[reflected limit|reflects]] [[finite limits]] in $\mathbf{E}^{op}$ from limits in $\mathbf{E}$, but finite limits in $\mathbf{E}^{op}$ are of course finite colimits in $\mathbf{E}$.

This elegant approach does involve a fair amount of categorical machinery (a [[monadicity theorem]], [[Beck-Chevalley conditions]], and consideration of up to six applications of the power object functor), making it a challenge to get across intuitively in say an undergraduate course.

Other approaches that are closer to naive or common sense set-theoretic reasoning are possible. In the case of quotient objects, to form the coequalizer of a parallel pair $f, g: X \rightrightarrows Y$, we outline a possible path to take (see also at \emph{\href{quotient+type#FromUnivalence}{quotient type -- from univalence}}):

\begin{itemize}%
\item Construct enough of the [[internal logic]] to make available logical operators $\wedge, \Rightarrow, \forall$.


\item Construct an internal [[intersection]] operator $\bigcap: P P X \to P X$ via the formula $\bigcap \Phi = \{x: X\; |\; \forall_{S: P X} \Phi \ni S \Rightarrow S \ni x\}$.


\item Construct the [[image]] of a map $f: X \to Y$ by taking the internal intersection of all [[subobjects]] of $Y$ through which $f$ factors.


\item To construct the [[coequalizer]] of $f, g: X \rightrightarrows Y$:

\begin{itemize}%
\item Take the image of $\langle f, g \rangle: X \to Y \times Y$ to get a relation $\langle p_1, p_2 \rangle: R \hookrightarrow Y \times Y$. According to Lemma \ref{image}, a coequalizer of $(p_1, p_2)$ is a coequalizer of $(f, g)$.
\item Then take the equivalence relation $\langle e_1, e_2 \rangle: E \hookrightarrow Y \times Y$ generated by $R$, by taking the intersection of all equivalence relations containing $R$; a coequalizer of $(e_1, e_2)$ is a coequalizer of $(p_1, p_2)$ (akin to the fact that any coequalizer must be the coequalizer of its kernel pair, as in Proposition \ref{galois}).
\item Take the image factorization of the map $\chi_E: Y \to P Y$ that classifies the relation $E \hookrightarrow Y \times Y$. That is, factor $\chi_E$ as an epi $q: Y \to Q$ followed by a mono $Q \to P Y$. Then $q$ is the desired coequalizer of $(e_1, e_2)$.

\end{itemize}


\end{itemize}
Notice that what these steps collectively do is form the object $Q$ of equivalence classes of the equivalence relation generated by the relation $f(x) \sim g(x)$, which is exactly what we would do in ordinary set theory. Full details will appear elsewhere.

\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

These notions have generalizations when $C$ is an [[(∞,1)-category]]:

\begin{itemize}%
\item an equivalence relation is then a [[groupoid object in an (∞,1)-category]]


\item it has an ``effective quotient'' if it is [[delooping|deloopable]].



\end{itemize}
For instance an [[action groupoid]] is a quotient of a group action in 2-category theory.

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

\begin{itemize}%
\item [[quotient set]]


\item [[quotient group]]


\item [[quotient module]]


\item [[quotient ring]]



\end{itemize}
([[quotient norm]])

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

\begin{itemize}%
\item [[fraction]]


\item [[subquotient]]


\item [[quotient space]], [[geometric invariant theory]]


\item [[quotient stack]]


\item In [[type theory]]/[[homotopy type theory]] the analogous concept is that of [[quotient types]].



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

\begin{itemize}%
\item [[Robert Paré]], \emph{Colimits in Topoi}, Bull. AMS 80 (1974) pp.556-561. (\href{http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13497-X/S0002-9904-1974-13497-X.pdf}{pdf})

\end{itemize}
[[!redirects quotient object]] [[!redirects quotient objects]] [[!redirects quotient]] [[!redirects quotients]]

[[!redirects regular quotient]] [[!redirects regular quotients]]

[[!redirects effective quotient]] [[!redirects effective quotients]]



\end{document}