Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
Both category theory and graph theory study patterns based on diagrams consisting of nodes and edges. Despite this surface impression, there is in fact very little interaction between the scientific communities of category theorists and of graph theorists.
This article is a modest bridge, indicating that the category of graphs (in the usual graph-theorist’s sense – see for example Diestel) has some very nice properties.
By a simple graph, we mean a set $V$ equipped with a symmetric reflexive relation $E$. Morphisms between simple graphs are functions which preserve the given relations. Elements of $V$ are called vertices. Write $E(x, y)$ to say $(x, y) \in E$.
When drawing graphs, we put an edge between vertices $x$, $y$ iff $x \neq y$ and $E(x, y)$. In particular, we will not bother drawing an edge between $x$ and itself; we can think of it as tacitly there if we like. Perhaps better yet, that tacit edge can be thought of as a path of length $0$ from $x$ to itself (and who bothers drawing a path of length $0$?). In any case, it is clear that there is a natural bijection between
simple graph structures on $V$ in the graph-theorist’s sense (i.e., graphs that are undirected, loop-free, and with at most one edge between two vertices), and
reflexive symmetric relations on $V$;
thus we are within our rights to identify these two concepts.
This maneuver allows a simple definition of morphism as a function (between the sets of vertices) that preserves the edge relation:
This then includes natural operations like edge contractions as quotients. Namely, if $f \colon G \to H$ is a morphism as defined above, and if we have $G(x, y)$ and $f(x) = f(y) = v$, we still have $H(f(x), f(y))$ by default, so we can think of $f$ as contracting the edge between $x$ and $y$ in $G$ down to an edge of length $0$ between $v$ and itself in $H$.
The category of simple graphs is denoted by $SimpGph$.
The category $SimpGph$ has very good properties. For example,
$SimpGph$ is a Grothendieck quasitopos. In particular, it is a regular category and even a logos. It is also $\infty$-extensive.
(See also Adamek and Herrlich.) Let $C$ be the category of sets (or cardinality) $1$ and $2$ and functions between them. A presheaf on $C$ is a directed reflexive (multi)graph equipped with a direction-reversing involution on the set of edges. For this presheaf topos, there is just one nontrivial $\neg\neg$-dense sieve, namely the inclusion
and so the category of $\neg\neg$-separated presheaves is equivalent to the category of presheaves $X$ such that the induced map
which is the source-target pairing $X(2) \to X(1) \times X(1)$, is monic. Such a structure is tantamount to a set $V = X(1)$ equipped with a reflexive symmetric binary relation. On the other hand, a Grothendieck quasitopos is, essentially by definition, the category of separated presheaves for a topology on a presheaf topos, in this case the $\neg\neg$-topology.
Being a quasitopos with small coproducts, it is $\infty$-extensive provided that coproducts are disjoint. However, this is trivial to check (it even suffices to check, according to Elephant 2.6.5, that $0 \to 1$ is a regular monomorphism, or that $1 + 1$ is a disjoint coproduct, which it obviously is).
It is easy to describe monos and epis in $SimpGph$. For, let $Vert \colon SimpGph \to Set$ be the underlying vertex-set forgetful functor. We have two easy results:
$Vert$ reflects monos and epis (because $Vert$ is faithful).
$Vert$ preserves limits and colimits (because it has a left adjoint $Disc$ and a right adjoint $Codisc$).
It follows that $Vert \colon SimpGph \to Set$ both preserves and reflects monos and epis. As a result, we can prove various simple exactness results in $SimpGph$. For instance:
In $SimpGph$, the pushout of any mono is a mono.
Suppose we have a pushout diagram in $SimpGph$:
Since $Vert$ preserves both pushouts and monos, and since the pushout of a mono in $Set$ is a mono, we have that $Vert(k)$ is monic in $Set$. Since $Vert$ reflects monos, this means $k$ is monic in $SimpGph$.
If $f, g \colon G \stackrel{\to}{\to} H$ are maps in $SimpGph$, then their equalizer $Eq(f, g) = i \colon K \to G$ is given at the vertex level by
and at the edge level by $K(x, y) \Leftrightarrow G(i(x), i(y))$; in other words, if $x, y$ belong to $K$ and there is an edge between them in $G$, then that edge lies in $K$. To express this last condition, we say the subgraph $i$ is full (at the edge level).
Thus, $i$ is a regular mono in $SimpGph$ iff both $Vert(i)$ is monic in $Set$ and $i$ is full at the edge level.
The coequalizer of $f$ and $g$, say $Coeq(f, g) = q \colon H \to Q$, is given at the vertex level by
and at the edge level by taking the image of the composite $E_H \hookrightarrow Vert(H)^2 \stackrel{q^2}{\to} Vert(Q)^2$, so that
Thus, $q$ is a regular epimorphism if it is surjective at both the vertex and edge levels.
The category of simple graphs has a ternary factorization system as follows: each morphism $f \colon G \to H$ factors as
where
$q$ is a surjection between vertex-sets and at the edge level, i.e., is a regular epi;
$a$ induces an identity between vertex-sets (hence is jointly monic and epic), but without necessarily being full at the edge level;
$i$ is given by an injection between vertex-sets and is full at the edge level, and (thus) is a regular mono.
The factorization of $f$ into $q$ followed by $i \circ a$ is the (regular epi)-mono factorization, while the factorization of $f$ into $a \circ q$ followed by $i$ is the epi-(regular mono) factorization. To round out the discussion, we prove that regular monos are stable under pushout (which ensures that the epi-(regular mono) factorization is an orthogonal factorization system).
In $SimpGph$, the pushout of any regular monomorphism along any map is a regular mono.
Let $j \colon G \to H$ be a regular mono in $SimpGph$, and let $f \colon G \to G'$ be any map. Form the pushout of the pair $(j, f)$ as a coequalizer diagram
where $i_1 \colon H \to H \sqcup G'$, $i_2 \colon G' \to H\sqcup G'$ are the coproduct inclusions. We are trying to show that $q \circ i_2$ is a regular mono. Certainly $Vert(q i_2)$ is monic in $Set$ (again since $Vert$ preserves monos and pushouts), so we just have to check that $q i_2$ is full at the edge level. So, suppose $H'(q i_2(x), q i_2 (y))$; we must show $G'(x, y)$. According to how we form coequalizers $q$, there exist $u, v \in H \sqcup G'$ such that $q(u) = q i_2(x)$, $q(v) = q i_2(y)$, and $(H \sqcup G')(u, v)$. Since $H$ and $G'$ are embedded disjointly in their coproduct, we have that
either both $u$ and $v$ belong to the $G'$-component, in which case $u = i_2(x)$ and $v = i_2(y)$ (but in that case we’re done because $G'(x, y)$ follows from $(H \sqcup G')(i_2 x, i_2 y)$, according to how the coproduct $H \sqcup G'$ is constructed);
or both $u$ and $v$ belong to the $H$-component. But according to how pushouts are constructed at the vertex level, this means there exist $g_1, g_2 \in G$ such that $f(g_1) = x$, $j(g_1) = u$ and $f(g_2) = y$, $j(g_2) = v$. Since $H(u, v)$, or $H(j(g_1), j(g_2))$, and $j$ is full at the edge level (because it is a regular mono), we have $G(g_1, g_2)$, and therefore $G'(f(g_1), f(g_2))$ or $G'(x, y)$, as desired.
$SimpGph$ is co-regular, i.e., $SimpGph^{op}$ is regular.
This is obvious from the first definition of “regular” given here.