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\section*{Quiv}

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

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

[[!include category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{as_a_topos}{As a topos}\dotfill \pageref*{as_a_topos} \linebreak
\noindent\hyperlink{Omega_graph}{The subobject classifier}\dotfill \pageref*{Omega_graph} \linebreak
\noindent\hyperlink{double_negation}{(Double) negation}\dotfill \pageref*{double_negation} \linebreak
\noindent\hyperlink{some_subcategories_and_adjunctions}{Some subcategories and adjunctions}\dotfill \pageref*{some_subcategories_and_adjunctions} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{$Quiv$} or \textbf{$DiGraph$} is the category of [[quivers]] or (as category theorists often call them) [[directed graphs]]. It can be viewed as the default categorical model for the concept of a category of graphs.

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

We can define a quiver to be a functor $G\colon X^{op} \to Set$, where $X^{op}$ is the [[category]] with an [[object]] $0$, an object $1$ and two [[morphisms]] $s, t\colon 1 \to 0$, along with [[identity morphisms]]. This lets us efficiently define $Quiv$ as the category of [[presheaves]] on $X$, where:

\begin{itemize}%
\item objects are [[functors]] $G\colon X^{op} \to Set$,
\item morphisms are [[natural transformations]] between such functors.

\end{itemize}
In other words, $Quiv$ is the [[functor category]] from this $X^{op}$ to [[Set]].

\hypertarget{as_a_topos}{}\subsection*{{As a topos}}\label{as_a_topos}

The category $Quiv = Set^{X^{op}}$, being a category of [[presheaves]], is a [[topos]]. The [[representable functors]] $X(-, 0), X(-, 1)$ may be pictured as the ``generic figures'' (generic vertex, generic edge) that occur in directed graphs:

\begin{displaymath}
X(-, 0) = \bullet, \qquad X(-, 1) = (x \stackrel{e}{\to} y)
\end{displaymath}
and from this picture we easily see that $X(-, 0)$ has two subobjects $\emptyset, \bullet$ whereas $X(-, 1)$ has five: $empty, x, y, (x, y), (x \stackrel{e}{\to} y)$.

Being a presheaf topos has a lot of nice consequences and instantly yields answer to questions like whether finite [[limit|limits]] of directed graphs exist or how to construct the [[exponential object|exponential]] quiver $Y^X$ of all homomorphisms $X\to Y$ between two quivers $X,Y$ in $Quiv$ since the answers are provided by topos theory.

In the following subsections some of the topos structure in $Quiv$ is worked out explicitly.

\hypertarget{Omega_graph}{}\subsubsection*{{The subobject classifier}}\label{Omega_graph}

Knowing the subobjects of the representable functors in turn allows us to calculate the structure of the [[subobject classifier]] $\Omega$ since they correspond to the elements of the value of $\Omega$ at the corresponding objects of $X^{op}$ i.e. the set of vertices of $\Omega$ is $\Omega(0)=\{\emptyset,\bullet\}$ and the set of edges is $\Omega(1)=\{empty, x, y, (x, y), (x \stackrel{e}{\to} y)\}$.

Whence pictured $\Omega$ is the quiver with two vertices and five edges that looks roughly like

\begin{displaymath}
\itexarray{
\emptyset & \underoverset{x}{y}{\rightleftarrows} & \bullet \\ 
\mathllap{empty} \circlearrowleft & & \circlearrowleft \circlearrowleft \mathrlap{x \stackrel{e}{\to} y} \\ 
 & & \mathllap{(x, y)}  
}
\end{displaymath}
(so there is one loop labeled ``empty'' at the vertex $\emptyset$, and two loops at the vertex $\bullet$, one labeled $(x, y)$ and the other $x \stackrel{e}{\to} y$).

How does $\Omega$ work? Suppose that $X\subseteq Y$ is a subgraph and $\chi_X:Y\to\Omega$ its characteristic map, then $\chi_X$ maps vertices of $Y$ not in $X$ to $\emptyset$ and vertices in $X$ to $\bullet$ (a vertex is either contained in a subgraph or not - the choice is binary and, accordingly, $\Omega$ needs two vertices to represent this). For edges the situation is more complicated since there are five ways (and, accordingly five edges in $\Omega$ to represent this) for an edge $z$ of $Y$ to be related to the subgraph $X$: the most straightforward is when $z$ has neither source nor target in $X$, such $z$ are definitely not in $X$ and are represented in $\Omega$ by the loop at $\emptyset$. Now suppose that $z$ has either source or target vertex in $X$ but not both: $\chi_X$ maps these to the maps $x,y$ between $\emptyset\rightleftarrows\bullet$, respectively. When $z$ has both source and target in $X$, the edge itself might or might not be in $X$, and the corresponding two cases are represented by the two loops at $\bullet$ , respectively, with $e$ representing the edges that are contained $X$.

\hypertarget{double_negation}{}\subsubsection*{{(Double) negation}}\label{double_negation}

The [[negation]] $\neg:\Omega\to \Omega$ is defined as the characteristic map of $\bot:1\to\Omega$. It specifies how

\begin{displaymath}
im(\bot)= \underoverset{{empty}}{\empty}{\circlearrowleft}
\end{displaymath}
sits as a subgraph in $\Omega$:

since $\bullet$ is not $im(\bot)$ whereas $\empty$ is, $\neg$ interchanges the two vertices and, accordingly, all loops at $\bullet$ must go ${empty}$ . Conversely, $empty$ goes to $e$ (since it is fully contained in $im(\bot)$). Now $x$ has its target but not its source in $im(\bot)$ hence it goes to $y$ whereas $y$ has its source but not its target in $im(\bot)$ and therefor goes to $x$.

Complementing a subobject $X\subseteq Y$ i.e. taking the subobject $\neg X$ of $Y$ that is classified by $\neg\circ\chi_X$ amounts to taking all vertices of $Y$ not in $X$ and all the edges in $Y$ between them.

Whence the result $\neg\neg X$ of applying $\neg$ twice to $X\subseteq Y$ amounts to adding to $X$ all the edges of $Y$ that have source and target in $X$. This implies in turn that a subgraph $X\subseteq Y$ is [[dense subobject|dense]] for the [[double negation|double negation topology]] $\neg\circ\neg:\Omega\to\Omega$ , precisely when it contains all vertices of $Y$ since complementing twice will throw into $\neg\neg X$ all the edges in $Y$ between all the vertices in $X$.

By definition, a quiver $X$ is [[separated object|separated]] for $\neg\neg$ when for every other quiver $Y$ and dense subobject $i:S\hookrightarrow Y$ and any map $f:S\to X$ there is at most one $g:Y\to X$ such that the following diagram commutes:

\begin{displaymath}
\itexarray{
S & & \\
i\downarrow &\searrow &f \\
Y &\underset{g}{\to} & X
}
\end{displaymath}
A separated quiver $X$ is a $\neg\neg$-sheaf when such a unique $g$ always exists.

Suppose that a quiver $X$ has a pair of parallel edges $w,z$. Then the subgraph $i:S\hookrightarrow X$ that is just like $X$ but has $w,z$ ommitted is dense in $X$. Let $\tau_{zw}:X\to X$ be the automorphism of $X$ that is just like the identity on $X$ except that it interchanges $w$ and $z$. Then $id_X\circ i=\tau_{zw}\circ i=i$ and one sees that $X$ is not separated.

Conversely, let $X$ be a quiver with at most one edge $x\to y$ between any pair $(x,y)$ of vertices and $f:S\to X$ be a map with $i:S\hookrightarrow Y$ is dense in $Y$. Since $i$ is a bijection on the vertex sets of $S$ and $Y$, if a factorization of $f$ through $g:Y\to X$ and $i$ exists the effect of $g$ on the vertices is uniquely determined by $f$ but since in $X$ there is at most one edge between any pair of vertices the image of any edge $a\to b$ in $Y$ under $g$ is already fixed: it is the unique edge between $g(a)$ and $g(b)$. In particular, one sees that a separated object $X$ is a sheaf precisely when there exists exactly one edge between any pair of vertices since then arbitrary edges in arbitrary factors $Y$ can be mapped to the appropriate edge in $X$. To sum up:

\begin{prop}
\label{negneg_subcats}\hypertarget{negneg_subcats}{}
A quiver $X$ is separated for the double negation topology $\neg\neg$ precisely if there exists at most an edge $a\to b$ between any pair $(a,b)$ of vertices. $X$ is a $\neg\neg$-sheaf precisely if there exists a unique edge $a\to b$ between any pair $(a,b)$. $\Box$

\end{prop}
The corresponding full subcategories are denoted by $Sep_{\neg\neg}(Quiv)$ and $Sh_{\neg\neg}(Quiv)$ , respectively. By generalities, it follows that $Sep_{\neg\neg}(Quiv)$ is a [[quasitopos]] and $Sh_{\neg\neg}(Quiv)$ is a [[Boolean topos]].

Quivers that have at most one edge between any pair of vertices can be called `simple' with the caveat that contrary to (the usual concept of) a simple graph they are allowed to have loops. Similarly, $\neg\neg$-sheaves can be called `complete'.

Since the edges of $\neg\neg$-separated quivers simply encode a binary endorelation on their vertex sets and being a morphism between $\neg\neg$-separated quivers then amounts to preserve that relation one sees that $Sep_{\neg\neg}(Quiv)$ and $Sh_{\neg\neg}(Quiv)$ are equivalent to the categories $Bin$ with objects $(X,\rho)$ where $X$ is a set and $\rho$ a binary endorelation on $X$, and, respectively, the category $TotalRel$ of sets equipped with the total relation. The latter can be identified with $Set$ since morphisms between sets equipped with the total relation behave just like ordinary functions between sets.

\hypertarget{some_subcategories_and_adjunctions}{}\subsubsection*{{Some subcategories and adjunctions}}\label{some_subcategories_and_adjunctions}

The inclusion $Sh_{\neg\neg}(Quiv)\hookrightarrow Sep_{\neg\neg}(Quiv)$ is actually an [[level|essential localisation]] since it corresponds (from the relational perspective) to the [[adjoint triple|adjoint string]] $e\dashv u\dashv t:Set\hookrightarrow Bin$ where $t$ maps a set $X$ to $(X,\tau_X)$ with $\tau_X$ the total relation on $X$, $u$ is the forgetful functor mapping $(X,\rho)$ to $X$ and, $e$ maps a set $X$ to $(X,\empty)$.

Similarly, $Sh_{\neg\neg}(Quiv)\overset{i}{\hookrightarrow} Quiv$ is an essential subtopos: if we identify sheavification $r$ with the functor that maps a quiver to the quiver on the same vertex set with edge set the total relation on the vertex set, then $l\dashv r\dashv i$ where $l$ forgets the edges of a complete quiver.

In particular, it follows then from general properties of the [[double negation|double negation topology]] that $Sh_{\neg\neg}(Quiv)$ is the [[Aufhebung]] of $0\dashv 1$. Whence, there exists indeed a notion of `codiscreteness' (= an Aufhebung of $0\dashv 1$) for quivers, namely `completeness', but it does not arise from a right adjoint to the [[section|section functor]] $\Gamma: Quiv\to Set$ that maps a quiver to its set of loops. Indeed, the adjoint string $\Pi\dashv\Delta\dashv\Gamma:Quiv\to Set$ that comes with the `discrete' inclusion $\Delta$ that maps a set to the quiver with vertex set $X$ and edge set precisely one loop for every vertex, is not a localisation since $\Pi\dashv\Delta$ is not a [[geometric morphism]] because $\Pi$ fails to preserve products.

Furthermore, since it is a general result for presheaf toposes (cf. La Palme Reyes et al. \hyperlink{RRZ04}{2004}, p.204) that $\Gamma$ has a right adjoint $B$ precisely if a generic figure has a point and in the case of Quiv neither the generic vertex nor the generic edge contains a loop, we see that the functor $\Gamma:Quiv\to Set$ has no right adjoint.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

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

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

\begin{itemize}%
\item R. T. Bumby, D. M. Latch, \emph{Categorical Constructions in Graph Theory} , Internat. J. Math. \& Math. Sci. \textbf{9} no.1 (1986) pp.1-16.


\item [[F. W. Lawvere]], \emph{Qualitative Distinctions between some Toposes of Generalized Graphs}, Cont. Math. \textbf{92} (1989) pp.261-299.


\item M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{Generic Figures and their Glueings}, Polimetrica Milano 2004.


\item S. Vigna, \emph{A Guided Tour in the Topos of Graphs} , arXiv.0306394 (2003). (\href{https://arxiv.org/abs/math/0306394}{abstract})



\end{itemize}
category: category

[[!redirects Quiv]] [[!redirects DiGraph]] [[!redirects Digraph]]



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