# nLab directed (n,r)-graph

Contents

### Context

#### Graph theory

graph theory

graph

category of simple graphs

# Contents

## Idea

The underlying higher directed graph of an $(n, r)$-category.

## Definition

### Directed (n,r)-pseudographs

For finite $r$, directed $(n,r)$-pseudographs are defined inductively as follows:

###### Definition

For $-2 \leq n \leq \infty$, a directed $(n,0)$-pseudograph is an ∞-groupoid that is n-truncated: an n-groupoid.

For $0 \leq r \lt \infty$, a directed $(n+1,r+1)$-pseudograph is an (n+1)-groupoid $G$ such that for every object $A$ and $B$, called a vertex, in $G$, there is a directed $(n,r)$-pseudograph of edges $Edge(A, B)$ between $A$ and $B$.

where $\infty + 1 = \infty$

For the case $r = \infty$, directed $(n, \infty)$-pseudographs are defined coinductively as follows:

###### Definition

For $-2 \leq n \leq \infty$, a directed $(n,\infty)$-pseudograph is an n-groupoid $G$ such that for every object $A$ and $B$, called a vertex, in $G$, there is a directed $(n,\infty)$-pseudograph of edges $Edge(A, B)$ between $A$ and $B$.

### Directed (n,r)-multigraphs

For finite $r$, directed $(n,r)$-multigraphs are defined inductively as follows:

###### Definition

For $-2 \leq n \leq \infty$, a directed $(n,0)$-multigraph is an ∞-groupoid that is n-truncated: an n-groupoid.

For $0 \leq r \lt \infty$, a directed $(n+1,r+1)$-multigraph is an (n+1)-groupoid $G$ such that for every object $A$ and $B$, called a vertex, in $G$, there is a directed $(n,r)$-multigraph of edges $Edge(A, B)$ between $A$ and $B$ such that given any vertex $A$, $Edge(A, A)$ is equivalent to the empty $\infty$-groupoid.

where $\infty + 1 = \infty$

For the case $r = \infty$, directed $(n, \infty)$-multigraphs are defined coinductively as follows:

###### Definition

For $-2 \leq n \leq \infty$, a directed $(n,\infty)$-multigraph is an n-groupoid $G$ such that for every object $A$ and $B$, called a vertex, in $G$, there is a directed $(n,\infty)$-multigraph of edges $Edge(A, B)$ between $A$ and $B$ such that given any vertex $A$, $Edge(A, A)$ is equivalent to the empty $\infty$-groupoid.

### Note on terminology

Some authors use “directed (n,r)-graph” to mean what we refer here as a “directed (n,r)-pseudograph”.

## The periodic table

### Directed $(n,r)$-pseudographs

There is a periodic table of directed $(n,r)$-pseudographs:

$r$↓\$n$$-2$$-1$$0$$1$$2$$\infty$
$0$trivialtruth valuesetgroupoid2-groupoid...infinity groupoid
$1$""directed loop graph?directed pseudographdirected (2,1)-pseudograph...directed (infinity,1)-pseudograph
$2$"""directed loop 2-graphdirected (2,2)-pseudograph...directed (infinity,2)-pseudograph
$3$""""directed loop 3-graph...directed (infinity,3)-pseudograph
$\infty$trivialtruth valuedirected loop graphdirected loop 2-graphdirected loop 3-graph...directed loop infinity-graph

### Directed $(n,r)$-multigraphs

There is a periodic table of directed $(n,r)$-multigraphs:

$r$↓\$n$$-2$$-1$$0$$1$$2$$\infty$
$0$trivialtruth valuesetgroupoid2-groupoid...infinity groupoid
$1$""directed graphdirected multigraph?directed (2,1)-multigraph...directed (infinity,1)-multigraph
$2$"""directed 2-graphdirected 2-multigraph...directed (infinity,2)-multigraph
$3$""""directed 3-graph...directed (infinity,3)-multigraph
$\infty$trivialtruth valuedirected graphdirected 2-graphdirected 3-graph...directed infinity-graph