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

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

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:

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 $B$.

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

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:

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.

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

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

$r$↓\$n$→ | $-2$ | $-1$ | $0$ | $1$ | $2$ | … | $\infty$ |
---|---|---|---|---|---|---|---|

$0$ | trivial | truth value | set | groupoid | 2-groupoid | ... | infinity groupoid |

$1$ | " | " | directed loop graph? | directed pseudograph | directed (2,1)-pseudograph | ... | directed (infinity,1)-pseudograph |

$2$ | " | " | " | directed loop 2-graph | directed (2,2)-pseudograph | ... | directed (infinity,2)-pseudograph |

$3$ | " | " | " | " | directed loop 3-graph | ... | directed (infinity,3)-pseudograph |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

$\infty$ | trivial | truth value | directed loop graph | directed loop 2-graph | directed loop 3-graph | ... | directed loop infinity-graph |

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

$r$↓\$n$→ | $-2$ | $-1$ | $0$ | $1$ | $2$ | … | $\infty$ |
---|---|---|---|---|---|---|---|

$0$ | trivial | truth value | set | groupoid | 2-groupoid | ... | infinity groupoid |

$1$ | " | " | directed graph | directed multigraph? | directed (2,1)-multigraph | ... | directed (infinity,1)-multigraph |

$2$ | " | " | " | directed 2-graph | directed 2-multigraph | ... | directed (infinity,2)-multigraph |

$3$ | " | " | " | " | directed 3-graph | ... | directed (infinity,3)-multigraph |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

$\infty$ | trivial | truth value | directed graph | directed 2-graph | directed 3-graph | ... | directed infinity-graph |

Last revised on May 16, 2022 at 17:46:57. See the history of this page for a list of all contributions to it.