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\section*{1-category equipped with relations}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{12categories_equipped_with_proarrows}{(1,2)-categories equipped with proarrows}\dotfill \pageref*{12categories_equipped_with_proarrows} \linebreak
\noindent\hyperlink{1categories_equipped_with_relations}{1-categories equipped with relations}\dotfill \pageref*{1categories_equipped_with_relations} \linebreak
\noindent\hyperlink{cartesian_1categories_equipped_with_relations}{Cartesian 1-categories equipped with relations}\dotfill \pageref*{cartesian_1categories_equipped_with_relations} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[2-category equipped with proarrows]] is a 2-category together with a 2-category of ``proarrows'' which are intended to generalize the arrows of $K$ in the same way that [[profunctors]] generalize the [[functors]] in [[Cat]]. Since profunctors are a [[categorification]] of [[relations]], it is natural to think of decategorifying such equipments to give a structure on a 1-category that equips it with ``relations''. We call this structure a \emph{1-category equipped with relations}.

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

\hypertarget{12categories_equipped_with_proarrows}{}\subsubsection*{{(1,2)-categories equipped with proarrows}}\label{12categories_equipped_with_proarrows}

Recall that a \emph{2-category equipped with proarrows} (aka ``proarrow equipment'' or ``equipment'') can be defined as a certain sort of [[double category]], with $\mathcal{V}(\underline{K}) = K$. If, in such a double category, any two squares with the same boundary are equal, we say that it is is a \textbf{(1,2)-category equipped with proarrows}, or a \textbf{(1,2)-category proarrow equipment}. This is equivalent to requiring that the 2-category of proarrows (and hence also the underlying 2-category of arrows) is [[locally posetal 2-category|locally posetal]], i.e. a (1,2)-category.

For example, if $V$ is any [[quantale]], then $V Cat$ is naturally a (1,2)-category equipped with proarrows. In particular, taking $V=\mathbb{2}$, we have a (1,2)-category proarrow equipment whose objects are [[preorders]].

\hypertarget{1categories_equipped_with_relations}{}\subsubsection*{{1-categories equipped with relations}}\label{1categories_equipped_with_relations}

A \textbf{1-category equipped with relations} is a (1,2)-category equipped with proarrows, regarded as a double category $\underline{K}$, together with an [[involution]] $\underline{K}^{h op} \cong \underline{K}$ which is (isomorphic to) the identity on objects and (vertical) arrows. Here $\underline{K}^{h op}$ denotes the horizontal opposite of a double category obtained by reversing the horizontal (pro-)arrows but not the vertical ones. We also call this structure a \textbf{relation equipment} or a \textbf{1-category proarrow equipment}.

In particular, the definition implies that we have an involution $K \cong K^{co}$ which is the identity on objects and arrows, which for a (1,2)-category means that $K$ is actually (equivalent to) a 1-category. Note though that the 2-category of proarrows (which we now call ``relations'') is still (like [[Rel]]) a (1,2)-category, not necessarily a 1-category.

For example, for any quantale $V$, the sub-2-category of $V Cat$ consisting of the \emph{symmetric} $V$-categories (those where $A(x,y) = A(y,x)$) is a 1-category equipped with relations. In particular, for $V=\mathbb{2}$, we have the relation equipment $\underline{Rel}$ of sets, functions, and binary relations.

In general, we can think of a relation equipment as generalizing some of the properties of $\underline{Rel}$. For instance, internal relations in any [[regular category]] also form a relation equipment.

\hypertarget{cartesian_1categories_equipped_with_relations}{}\subsection*{{Cartesian 1-categories equipped with relations}}\label{cartesian_1categories_equipped_with_relations}

It is proven in

\begin{itemize}%
\item Carboni, Kelly, Wood, ``A 2-categorical approach to change of base and geometric morphisms, I'' (\href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_1/CTGDC_1991__32_1_47_0/CTGDC_1991__32_1_47_0.pdf}{PDF})

\end{itemize}
that a (1,2)-category is a [[cartesian bicategory]] precisely when it is a [[cartesian object]] in a suitable 2-category of proarrow equipments (where we make a bicategory $M$ into an equipment by taking the proarrows to be those of $M$ and the arrows to be the ``maps'' in $M$, i.e. the morphisms having right adjoints). Here is a rough sketch of the argument, using the double-category description of equipments.

\begin{utheorem}
Let $\underline{K}$ be a 1-category equipped with relations, which is a [[cartesian object]] in the 2-category of relation equipments (that is, it is a \textbf{cartesian relation equipment}). Then $\mathcal{H}(\underline{K})$ is a cartesian bicategory.

\end{utheorem}
\begin{proof}
That $\underline{K}$ is a cartesian object means, in particular, that it is a [[pseudomonoid]] in the 2-category of equipments. By lifting the coherence data from arrows to representable proarrows, it follows that $\mathcal{H}(\underline{K})$ is a monoidal 2-category. Being a cartesian object also gives a cartesian product on objects and proarrows, with diagonals $\Delta\colon X\to X\times X$, and lifting these arrows to representable proarrows $\Delta_\bullet$ and $\Delta^\bullet$ gives each object a commutative monoid and comonoid structure. Now for any proarrow $\phi\colon X\to Y$, the square

\begin{displaymath}
\itexarray{X & \overset{\phi}{\to} & Y\\
  ^\Delta \downarrow & \Downarrow & \downarrow^\Delta\\
  X\times X& \underset{\phi\times \phi}{\to} & Y\times Y}
\end{displaymath}
in $\underline{K}$ induces 2-cells, i.e. inequalities, $\Delta_\bullet \phi \le (\phi\times\phi)\Delta_\bullet$ and $\phi \Delta^\bullet  \le \Delta^\bullet(\phi\times\phi)$.

\end{proof}
A [[bicategory of relations]] is a (1,2)-category which is a cartesian bicategory, and which also satisfies some additional conditions. We can also construct this structure starting from a relation equipment.

\begin{utheorem}
Let $\underline{K}$ be a relation equipment satisfying the hypotheses of the previous theorem, and suppose in addition that every proarrow $\phi\colon x\nrightarrow y$ in $\underline{K}$ can be written as $f_\bullet g^\bullet$ for some (vertical) arrows $f$ and $g$. (That is, ``tabulations'' in a certain sense exist.) Then $\mathcal{H}(\underline{K})$ is a [[bicategory of relations]].

\end{utheorem}
\begin{proof}
We first verify the axiom $\Delta^\bullet \Delta_\bullet = 1$. Since $\Delta^\bullet \Delta_\bullet$ is the restriction of $1_{X\times X}$ along $\Delta$ on both sides, it suffices to show that

\begin{displaymath}
\itexarray{X & \overset{1_X}{\to} & X\\
  ^\Delta\downarrow &\Downarrow& \downarrow^\Delta\\
  X\times X& \underset{1_{X\times X}}{\to} & X\times X}
\end{displaymath}
is a cartesian 2-cell in $\underline{K}$. But if we have any other square

\begin{displaymath}
\itexarray{A & \overset{\phi}{\to} & B\\
  ^{(f,g)}\downarrow &\Downarrow& \downarrow^{(h,k)}\\
  X\times X& \underset{1_{X\times X}}{\to} & X\times X}
\end{displaymath}
then $(f,g)$ factoring through $\Delta$ means that $f=g$, and likewise $h=k$. Composing the given square with the projection

\begin{displaymath}
\itexarray{X\times X & \overset{1_{X\times X}}{\to} & X\times X\\
  \downarrow &\Downarrow & \downarrow\\
  X& \underset{1_X}{\to} & X}
\end{displaymath}
(which comes from being a cartesian object in $Equipments$), we obtain a square

\begin{displaymath}
\itexarray{A & \overset{\phi}{\to} & B \\
  ^f\downarrow &\Downarrow & \downarrow^g\\
  X& \underset{1_X}{\to} & X}
\end{displaymath}
which factors the given square through the putative cartesian one. The factorization is unique since all 2-cells are unique.

We now verify the Frobenius axiom $\Delta^\bullet \Delta_\bullet = (1\times \Delta_\bullet)(\Delta^\bullet \times 1)$. Since $\Delta$ is associative, we have a square

\begin{displaymath}
\itexarray{X & \overset{1_X}{\to} & X\\
  ^\Delta\downarrow && \downarrow^\Delta\\
  X\times X & \Downarrow & X\times X\\
  ^{1\times \Delta}\downarrow && \downarrow^{\Delta\times 1}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X}
\end{displaymath}
and therefore a square

\begin{displaymath}
\itexarray{X\times X & \overset{\Delta^\bullet \Delta_\bullet}{\to} & X\times  X\\
  ^{1\times \Delta}\downarrow && \downarrow^{\Delta\times 1}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X}
\end{displaymath}
and it suffices to show that this is a cartesian 2-cell. So suppose given a square

\begin{displaymath}
\itexarray{A & \overset{\phi}{\to} & B\\
  ^{(f,g,g)}\downarrow & \Downarrow & \downarrow^{(h,h,k)}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X.}
\end{displaymath}
The fact that $g$ and $h$ appear twice is equivalent to saying that the left and right boundaries of this square factor through $1\times\Delta$ and $\Delta\times 1$, respectively. Now by assumption, $\phi = u_\bullet v^\bullet$ for some $u\colon C\to B$ and $v\colon C\to A$. Thus our square is equivalent to one

\begin{displaymath}
\itexarray{C & \overset{1_C}{\to} & C\\
  ^{(f v,g v,g v)}\downarrow & \Downarrow & \downarrow^{(h u,h u,k u)}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X.}
\end{displaymath}
But this is just a 2-cell in the vertical category $K$, which is a 1-category; hence we have $(f v,g v, g v) = (h u, h u, k u)$ and thus $f v = h u = g v = k u$. Calling their common value $m$, we thus have a composite square

\begin{displaymath}
\itexarray{C & = & C & = & C\\
  ^{(m,m)}\downarrow && \downarrow^{m} && \downarrow^{(m,m)}\\
  X\times X & \underset{\Delta^\bullet }{\to} & X & \underset{\Delta_\bullet}{\to} & X\times X}
\end{displaymath}
(since $\Delta m = (m,m)$) which gives us the desired factorization. The other Frobenius axiom is, of course, dual.

\end{proof}
\begin{ucorollary}
If $\underline{K}$ is a relation equipment satisfying the hypotheses of the theorem, then $\mathcal{H}(\underline{K})$ is an [[allegory]].

\end{ucorollary}
\begin{proof}
It is shown \href{http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html}{here} that any bicategory of relations is an allegory.

\end{proof}
\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

Other attempted axiomatizations of the same idea ``something that acts like the category of relations in a regular category'' include:

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


\item [[bicategory of relations]]



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

A comparison of ``regular proarrow equipments'' with ``regular [[fibrations]] of [[subobjects]]'' is in

\begin{itemize}%
\item [[Finn Lawler]], \emph{Fibrations of predicates and bicategories of relations}, \href{http://arxiv.org/abs/1502.08017}{arXiv}

\end{itemize}
[[!redirects (1,2)-category equipment]] [[!redirects (1,2)-category equipped with proarrows]] [[!redirects 1-category equipped with proarrows]] [[!redirects relation equipment]] [[!redirects relation equipments]] [[!redirects 1-category relation equipment]] [[!redirects 1-category equipment]] [[!redirects 1-category equipments]] [[!redirects 1-category proarrow equipment]] [[!redirects (1,2)-category proarrow equipment]]



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