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\begin{document}

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\section*{congruence}

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

\hypertarget{relations}{}\paragraph*{{Relations}}\label{relations}

[[!include relations - contents]]

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

[[!include category theory - contents]]

\hypertarget{congruences}{}\section*{{Congruences}}\label{congruences}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak


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

\begin{defn}
\label{Definition}\hypertarget{Definition}{}
In a [[finitely complete category]] $C$, a \textbf{congruence} on an object $X$ is an [[internalization|internal]] [[equivalence relation]] on $X$.

This means that it consists of a [[subobject]] of the product $R \subseteq X \times X$ equipped with the following [[morphisms]]:

\begin{itemize}%
\item internal [[reflexive relation|reflexivity]]: $r \colon X \to R$ which is a [[section]] both of $p_1$ and of $p_2$;


\item internal [[symmetric relation|symmetry]]: $s \colon R \to R$ which interchanges $p_1$ and $p_2$, namely $p_1\circ s = p_2$ and $p_2\circ s = p_1$;


\item internal [[transitive relation|transitivity]]: $t: R \times_X R \to R$; where with the notation for the projections in the [[cartesian square]]

\begin{displaymath}
\itexarray{
  R \times_X R & \stackrel{q_2}\rightarrow & R
  \\
  \downarrow^{\mathrlap{q_1}} && \downarrow^{\mathrlap{p_1}}
  \\
  R & \stackrel{p_2}\rightarrow & X
}
\end{displaymath}
the following holds: $p_1 = \pi_1 \circ i$, $p_2 = \pi_2 \circ i$, $p_1\circ q_1 = p_1\circ t$ and $p_2\circ q_2 = p_2\circ t$.

\begin{displaymath}
R\stackrel{i}\hookrightarrow X \times X \stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}} X
\end{displaymath}


\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Since $i$ is a [[monomorphism]], the maps $r$, $s$, and $t$ are necessarily unique if they exist.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Equivalently, a congruence on $X$ is an [[internal category]] with $X$ the object of [[objects]], such that the (source,target)-map is a [[monomorphism]] and such that if there is there is a morphism $x_1 \to x_2$ then there is also a morphism $x_2 \to x_1$ (internally).

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
We can equivalently define a congruence $R$ as (a representing object of) a representable sub-presheaf of $\hom(-, X \times X)$ so that for each object $Y$, the composite of $R(Y) \hookrightarrow \hom(Y, X \times X) \cong \hom(Y, X) \times \hom(Y, X)$ exhibits $R(Y)$ as an equivalence relation on the set $\hom(Y, X)$. The upshot of this definition is that it makes sense even when $C$ is not finitely complete.

\end{remark}
\begin{defn}
\label{EffectiveCongruence}\hypertarget{EffectiveCongruence}{}
A congruence which is the kernel pair of some morphism (example \ref{KernelPairIsCongruence}) is called \textbf{effective}.

\end{defn}
\begin{defn}
\label{QuotientObject}\hypertarget{QuotientObject}{}
The [[coequalizer]] of a congruence is called a \textbf{[[quotient object]]}.

The quotient of an effective congruence is called an \textbf{effective quotient}.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A [[regular category]] is called an [[exact category]] if every congruence is effective.

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

\begin{prop}
\label{}\hypertarget{}{}
An effective congruence, def. \ref{EffectiveCongruence}, is always the [[kernel pair]] of its quotient, def. \ref{QuotientObject}, if that quotient exists.

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

\begin{example}
\label{KernelPairIsCongruence}\hypertarget{KernelPairIsCongruence}{}
Every [[kernel pair]] is a congruence.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
An [[equivalence relation]] is precisely a congruence in [[Set]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The eponymous example is congruence modulo $n$ (for a fixed [[natural number]] $n$), which can be considered a congruence on $\mathbb{N}$ in the category of [[rigs]], or on $\mathbb{Z}$ in the category of [[rings]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
A [[quotient group]] by a [[normal subgroup]] $K \hookrightarrow G$ is the quotient of the relation $G \times K \stackrel{(p_1,p_2)}{\hookrightarrow} G \times G$, where $p_1$ is projection on the first factor and $p_2$ is multiplication in $G$ (these are source and target maps in the [[action groupoid]] $G \sslash K$).

A special case of this is that of a \emph{[[quotient module]]}.

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

\begin{itemize}%
\item The notions of [[regular category]] and [[exact category]] can naturally be formulated in terms of congruences. A ``higher arity'' version, corresponding to [[coherent categories]] and [[pretoposes]] is discussed at [[familial regularity and exactness]].


\item A [[Mal'cev category]] is a [[finitely complete category]] in which every internal relation satisfying reflexivity is thereby actually a congruence.


\item [[higher category theory|Higher-categorical]] generalizations are that of a [[2-congruence]] and of a [[groupoid object in an (∞,1)-category]]. See also [[(n,r)-congruence]].



\end{itemize}
[[!redirects congruence]] [[!redirects congruences]] [[!redirects congruence relation]] [[!redirects congruence relations]] [[!redirects internal equivalence relation]] [[!redirects internal equivalence relations]] [[!redirects internal congruence]] [[!redirects internal congruences]]



\end{document}