Example 8.1.2.1.4 (00WQ): Ordinal Categories

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

The $n$th ordinal category is the category $\mathbb {n}$ where ^[1]

Objects. We have

\[ \text{Obj}\webleft (\mathbb {n}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft [0\webright ],\ldots ,\webleft [n\webright ]\webright\} . \]
Morphisms. For each $\webleft [i\webright ],\webleft [j\webright ]\in \text{Obj}\webleft (\mathbb {n}\webright )$, we have

\[ \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [j\webright ]\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \webleft\{ \text{id}_{\webleft [i\webright ]}\webright\} & \text{if $\webleft [i\webright ]=\webleft [j\webright ]$,}\\ \webleft\{ \webleft [i\webright ]\to \webleft [j\webright ]\webright\} & \text{if $\webleft [j\webright ]<\webleft [i\webright ]$,}\\ \emptyset & \text{if $\webleft [j\webright ]>\webleft [i\webright ]$.} \end{cases} \]
Identities. For each $\webleft [i\webright ]\in \text{Obj}\webleft (\mathbb {n}\webright )$, the unit map

\[ \mathbb {1}^{\mathbb {n}}_{\webleft [i\webright ]} \colon \text{pt}\to \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [i\webright ]\webright ) \]

of $\mathbb {n}$ at $\webleft [i\webright ]$ is defined by

\[ \text{id}^{\mathbb {n}}_{\webleft [i\webright ]} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\webleft [i\webright ]}. \]
Composition. For each $\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]\in \text{Obj}\webleft (\mathbb {n}\webright )$, the composition map

\[ \circ ^{\mathbb {n}}_{\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]} \colon \textup{Hom}_{\mathbb {n}}\webleft (\webleft [j\webright ],\webleft [k\webright ]\webright ) \times \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [j\webright ]\webright ) \to \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [k\webright ]\webright ) \]

of $\mathbb {n}$ at $\webleft (\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]\webright )$ is defined by

\[ \begin{gathered} \text{id}_{\webleft [i\webright ]}\circ \text{id}_{\webleft [i\webright ]} = \text{id}_{\webleft [i\webright ]},\\ \webleft (\webleft [j\webright ]\to \webleft [k\webright ]\webright )\circ \webleft (\webleft [i\webright ]\to \webleft [j\webright ]\webright ) = \webleft (\webleft [i\webright ]\to \webleft [k\webright ]\webright ).\end{gathered} \]

Footnotes

[1] In other words, $\mathbb {n}$ is the category associated to the poset

\[ \webleft [0\webright ]\to \webleft [1\webright ]\to \cdots \to \webleft [n-1\webright ]\to \webleft [n\webright ]. \]

The category $\mathbb {n}$ for $n\geq 2$ may also be defined in terms of $\mathbb {0}$ and joins (

): we have isomorphisms of categories

\begin{align*} \mathbb {1} & \cong \mathbb {0}\star \mathbb {0},\\ \mathbb {2} & \cong \mathbb {1}\star \mathbb {0}\\ & \cong \webleft (\mathbb {0}\star \mathbb {0}\webright )\star \mathbb {0},\\ \mathbb {3} & \cong \mathbb {2}\star \mathbb {0}\\ & \cong \webleft (\mathbb {1}\star \mathbb {0}\webright )\star \mathbb {0}\\ & \cong \webleft (\webleft (\mathbb {0}\star \mathbb {0}\webright )\star \mathbb {0}\webright )\star \mathbb {0},\\ \mathbb {4} & \cong \mathbb {3}\star \mathbb {0}\\ & \cong \webleft (\mathbb {2}\star \mathbb {0}\webright )\star \mathbb {0}\\ & \cong \webleft (\webleft (\mathbb {1}\star \mathbb {0}\webright )\star \mathbb {0}\webright )\star \mathbb {0}\\ & \cong \webleft (\webleft (\webleft (\mathbb {0}\star \mathbb {0}\webright )\star \mathbb {0}\webright )\star \mathbb {0}\webright )\star \mathbb {0}, \end{align*}

and so on.