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\begin{document}

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\section*{relation}

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

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

[[!include relations - contents]]

\hypertarget{relations_2}{}\section*{{Relations}}\label{relations_2}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{DefinitionGeneralCase}{General case}\dotfill \pageref*{DefinitionGeneralCase} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{morphisms}{Morphisms}\dotfill \pageref*{morphisms} \linebreak
\noindent\hyperlink{binary_relations}{Binary relations}\dotfill \pageref*{binary_relations} \linebreak
\noindent\hyperlink{kinds_of_binary_relations}{Kinds of binary relations}\dotfill \pageref*{kinds_of_binary_relations} \linebreak
\noindent\hyperlink{The2PosetOfBinaryRelations}{The $2$-poset of binary relations}\dotfill \pageref*{The2PosetOfBinaryRelations} \linebreak
\noindent\hyperlink{endorel}{The quasitopos of endorelations}\dotfill \pageref*{endorel} \linebreak
\noindent\hyperlink{relation_closures_as_reflexive_subcategories_of_}{Relation closures as reflexive subcategories of $EndoRel$}\dotfill \pageref*{relation_closures_as_reflexive_subcategories_of_} \linebreak
\noindent\hyperlink{Generalization}{Generalisation}\dotfill \pageref*{Generalization} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \emph{relation} is the extension of a \emph{[[predicate]]}. That is, if you have a statement whose [[truth value]] may depend on some [[variables]], then you get a relation that consists of those instantiations of the variables that make the statement [[true]]. Equivalently, you can think of a relation as a [[function]] whose [[target]] is the set of truth values.

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

\hypertarget{DefinitionGeneralCase}{}\subsubsection*{{General case}}\label{DefinitionGeneralCase}

\begin{defn}
\label{}\hypertarget{}{}
Given a [[family of sets|family]] $(A_i)_{i: I}$ of [[sets]], a \textbf{relation} on that family is a [[subset]] $R$ of the [[cartesian product]] $\prod_{i: I} A_i$.

Equivalently, this is a [[function]] from $\prod_{i: I} A_i$ to the set $\TV$ of [[truth values]] (because $\TV$ is the [[subobject classifier]] in [[Set]]).

Also equivalently, this is a [[monomorphism]] in/[[subobject]] of the [[cartesian product]]

\begin{displaymath}
R \hookrightarrow \prod_{i \colon I} A_i
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $I$ the 2-element set (a \emph{binary relation}) this is in particular a binary [[correspondence]]

\begin{displaymath}
\itexarray{
     && R
     \\
     & \swarrow && \searrow
     \\
    A_1 && && A_1
  }
  \,.
\end{displaymath}
Hence relations are precisely the [[(-1)-truncated]] [[correspondences]].

This identification induces a natural notion of [[composition]] of relations from the composition in the [[category of correspondences]], see below at \emph{\hyperlink{The2PosetOfBinaryRelations}{The 2-poset of binary relations}}.

Formulated this way, the notion of relation generalizes to [[categories]] other than [[Set]] and to [[higher category theory]]. For more on this see at \emph{\hyperlink{Generalization}{Generalizations}} below.

\end{remark}
\hypertarget{special_cases}{}\subsubsection*{{Special cases}}\label{special_cases}

A \textbf{nullary relation} is a relation on the empty family of sets. This is the same as a [[truth value]].

A \textbf{unary relation on $A$} is a relation on the singleton family $(A)$. This is the same as a [[subset]] of $A$.

A \textbf{binary relation on $A$ and $B$} is a relation on the family $(A,B)$, that is a subset of $A \times B$. This is also called a \textbf{relation from $A$ to $B$}, especially in the context of the $2$-category [[Rel]] described below, or sometimes called a \textbf{heterogenous relation}.

A \textbf{binary relation on $A$} is a relation on $(A,A)$, that is a relation from $A$ to itself. This is sometimes called a \textbf{homogenous relation} on $A$, simply a \textbf{relation on $A$}, or just an \textbf{endorelation} with its set implicit as a property if not explicitly mentioned.

An \textbf{$n$-ary relation} on $A$ is a relation on a family of $n$ copies of $A$, that is a subset of $A^n$.

For a binary relation, one often uses a symbol such as $\sim$ and writes $a \sim b$ instead of $(a,b) \in \sim$. Actually, even when a relation is given by a letter such as $R$, one often sees $a R b$ instead of $(a,b) \in R$, although now that does not look so good.

\hypertarget{morphisms}{}\subsection*{{Morphisms}}\label{morphisms}

If $A$ and $B$ are each sets equipped with a relation, then what makes a [[function]] $f: A \to B$ a \emph{morphism} of sets so equipped?

There are really two ways to do this, shown below. (We will write these as if each set is equipped with a binary relation $\sim$, but any fixed arity would work.)

\begin{itemize}%
\item $f$ \textbf{preserves} the relation if $x \sim y \;\Rightarrow\; f(x) \sim f(y)$ always;
\item \emph{f} \textbf{reflects} the relation if $x \sim y \;\Leftarrow\; f(x) \sim f(y)$ always.

\end{itemize}
Now, if $f$ is a [[bijection]], then it preserves the relation if and only if its inverse reflects it, so clearly an [[isomorphism]] of relation-equipped sets should do both. What about a mere morphism?

In general, it's more natural to require only preservation; these are the morphisms you get if you consider a set with a relation as a models of a [[algebraic theory|finite-limit theory]] or a simply [[directed graph]].

But in some contexts, particularly when dealing only with [[irreflexive relation]]s, we instead require (only) that a morphism reflect the relation. Sometimes an even stricter condition is imposed, as for [[well-order]]s. But even in these cases, the definition of isomorphism comes out the same.

\hypertarget{binary_relations}{}\subsection*{{Binary relations}}\label{binary_relations}

Binary relations are especially widely used.

\hypertarget{kinds_of_binary_relations}{}\subsubsection*{{Kinds of binary relations}}\label{kinds_of_binary_relations}

Special kinds of relations from $A$ to $B$ include:

\begin{itemize}%
\item [[functional relations]],
\item [[entire relations]],
\item [[one-to-one relation]]s,
\item [[onto relation]]s.

\end{itemize}
Combinations of the above properties of binary relations produce:

\begin{itemize}%
\item [[functions]],
\item [[partial functions]],
\item [[injections]],
\item [[surjections]],
\item [[bijections]].

\end{itemize}
Special kinds of binary relations on $A$ (so from $A$ to itself) additionally include:

\begin{itemize}%
\item [[reflexive relation|reflexive]] and [[irreflexive relation|irreflexive]] relations;
\item [[symmetric relation|symmetric]], [[antisymmetric relation|antisymmetric]], and [[asymmetric relation|asymmetric]] relations;
\item [[transitive relations]] and [[comparisons]];
\item left and right [[euclidean relations]];
\item [[total relation|total]] and [[connected relation|connected]] relations;
\item [[extensional relation|extensional]] and [[well-founded relation|well-founded]] relations.

\end{itemize}
Combinations of the above properties of binary relations produce:

\begin{itemize}%
\item [[equivalence relations]],
\item [[partial equivalence relations]],
\item [[apartness relations]],
\item the various kinds of [[orders]] (which see).

\end{itemize}
\hypertarget{The2PosetOfBinaryRelations}{}\subsubsection*{{The $2$-poset of binary relations}}\label{The2PosetOfBinaryRelations}

Binary relations form a $2$-[[2-category|category]] (in fact a $2$-[[2-poset|poset]]) [[Rel]], which is the basic example of an [[allegory]].

The [[objects]] are [[sets]], the [[morphisms]] from $A$ to $B$ are the binary relations on $A$ and $B$, and there is a [[2-morphism]] from $R$ to $S$ (both from $A$ to $B$) if $R$ implies $S$ (that is, when $(a,b) \in R$, then $(a,b) \in S$).

The interesting definition is [[composition]]

\begin{defn}
\label{CompositionOfRelations}\hypertarget{CompositionOfRelations}{}
If $R$ is a relation from $A$ to $B$ and $S$ is a relation from $B$ to $C$, then their \emph{composite relation} -- written $S \circ R$ or $R;S$ -- from $A$ to $C$ is defined as follows:

\begin{displaymath}
(a,c) \in R;S \;\Leftrightarrow\; \exists b: B,\; (a,b) \in R \;\wedge\; (b,c) \in S.
\end{displaymath}
The identity morphism is given by [[equality]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[composition]] operation of relation from def. \ref{CompositionOfRelations} is induced by the composition of the underlying [[correspondences]], followed by [[(-1)-truncated|(-1)-truncation]].

\end{remark}
The special properties of the kinds of binary relations listed earlier can all be described in terms internal to [[Rel]]; most of them make sense in any [[allegory]]. Irreflexive and asymmetric relations are most useful if the allegory's [[hom-object|hom-poset]]s have [[bottom]] elements, and linear relations require this. Comparisons require the hom-posets to have finite [[union|unions]], and well-founded relations require some sort of higher-order structure.

As a [[function]] may be seen as a functional, entire relation, so the category [[Set]] of sets and functions is a [[subcategory]] of [[Rel]] (in fact a [[replete subcategory|replete]] and locally [[full subcategory|full]] sub-$2$-category).

\hypertarget{endorel}{}\subsubsection*{{The quasitopos of endorelations}}\label{endorel}

Endorelations on sets are the objects of the [[quasitopos]] \textbf{$EndoRel$} or \textbf{$Bin$}. It is a [[reflective subcategory]] of [[Quiv]] the [[category of presheaves|presheaf topos]] of quivers and its morphisms are quiver morphisms. Endorelations are the [[separated presheaf|separated presheaves]] for the [[double negation\#in\_topos\_theory|double negation topology]] on $Quiv$. ``Separated'' here translates to a quiver having at most one arc between pairs of verticies. The [[reflective subcategory|reflector]] $Quiv \to EndoRel$ collapses parallel arcs together. Such quivers might also be called \textbf{singular} or \textbf{simple} though sometimes ``simple'' also means ``no loops''.

\hypertarget{relation_closures_as_reflexive_subcategories_of_}{}\paragraph*{{Relation closures as reflexive subcategories of $EndoRel$}}\label{relation_closures_as_reflexive_subcategories_of_}

All of the sub-types of endorelations with positive conditions ([[reflexive relation|reflexive]], [[symmetric relation|symmetric]], [[transitive relations|transitive]], and left and right [[euclidean relations|euclidean]]) and their combinations have an associated [[Moore closure|closure]] that can produce one from an arbitrary relation. Such a closure [[completion|completes]] a relation by adding the least number of arcs such that the conditions are satisfied. Within $EndoRel$ these closures are reflectors that produce reflective subcategories. For example the \textbf{symmetric closure} $sym: EndoRel \to EndoSym$ will (possibly) enlarge a quiver that contains any arc $v_a \to v_b$ to one that also contains $v_b \to v_a$. The \textbf{transitive and reflexive closure} $transRef: EndoRel \to EndoTransRef$ produces a category which is isomorphic to [[preorder|PreOrd]] though its objects are the underlying quivers of the preorders which are the objects of $PreOrd$.

In addition to being reflective, the categories from the \emph{symmetric}, \emph{reflexive}, and \emph{symmetric and reflexive} closures are also quasitoposes that can be [[quasitopos\#exampsep|directly defined]] through double negation separation. The \emph{symmetric and reflexive} closure ([[category of simple graphs|SimpGph]]) is also a [[quasitopos|Grothendieck quasitopos]]. On the other hand $PreOrd$ is not a quasitopos because it is not a [[regular category]].

\hypertarget{Generalization}{}\subsection*{{Generalisation}}\label{Generalization}

Most of the preceding makes sense in any [[category]] with enough [[products]]; giving rise to [[internal relation|internal relations]], for instance \emph{[[congruences]]} in the case of internal equivalence relations.

Probably the trickiest bit is the definition of [[composition]] of binary relations, so not every category with [[finite products]] has an [[allegory]] of relations. In fact, in a certain precise sense, a category has an allegory of relations if and only if it is [[regular category|regular]]. It can then be recovered from this allegory by looking at the functional entire relations.

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

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


\item [[logical relation]]



\end{itemize}
[[!redirects relation]] [[!redirects relations]] [[!redirects relation theory]] [[!redirects theory of relations]]

[[!redirects nullary relation]] [[!redirects nullary relations]]

[[!redirects unary relation]] [[!redirects unary relations]]

[[!redirects binary relation]] [[!redirects binary relations]]



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