The $n$th ordinal category is the category $\mathbb {n}$ where1
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Objects. We have
\[ \operatorname {\mathrm{Obj}}\webleft (\mathbb {n}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \webleft [0\webright ],\ldots ,\webleft [n\webright ]\right\} . \] -
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Morphisms. For each $\webleft [i\webright ],\webleft [j\webright ]\in \operatorname {\mathrm{Obj}}\webleft (\mathbb {n}\webright )$, we have
\[ \operatorname {\mathrm{Hom}}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [j\webright ]\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \left\{ \operatorname {\mathrm{id}}_{\webleft [i\webright ]}\right\} & \text{if $\webleft [i\webright ]=\webleft [j\webright ]$,}\\ \left\{ \webleft [i\webright ]\to \webleft [j\webright ]\right\} & \text{if $\webleft [j\webright ]<\webleft [i\webright ]$,}\\ \text{Ø}& \text{if $\webleft [j\webright ]>\webleft [i\webright ]$.} \end{cases} \] -
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Identities. For each $\webleft [i\webright ]\in \operatorname {\mathrm{Obj}}\webleft (\mathbb {n}\webright )$, the unit map
\[ \mathbb {1}^{\mathbb {n}}_{\webleft [i\webright ]} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [i\webright ]\webright ) \]of $\mathbb {n}$ at $\webleft [i\webright ]$ is defined by
\[ \operatorname {\mathrm{id}}^{\mathbb {n}}_{\webleft [i\webright ]} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{\webleft [i\webright ]}. \] -
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Composition. For each $\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]\in \operatorname {\mathrm{Obj}}\webleft (\mathbb {n}\webright )$, the composition map
\[ \circ ^{\mathbb {n}}_{\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]} \colon \operatorname {\mathrm{Hom}}_{\mathbb {n}}\webleft (\webleft [j\webright ],\webleft [k\webright ]\webright ) \times \operatorname {\mathrm{Hom}}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [j\webright ]\webright ) \to \operatorname {\mathrm{Hom}}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [k\webright ]\webright ) \]of $\mathbb {n}$ at $\webleft (\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]\webright )$ is defined by
\[ \begin{gathered} \operatorname {\mathrm{id}}_{\webleft [i\webright ]}\circ \operatorname {\mathrm{id}}_{\webleft [i\webright ]} = \operatorname {\mathrm{id}}_{\webleft [i\webright ]},\\ \webleft (\webleft [j\webright ]\to \webleft [k\webright ]\webright )\circ \webleft (\webleft [i\webright ]\to \webleft [j\webright ]\webright ) = \webleft (\webleft [i\webright ]\to \webleft [k\webright ]\webright ).\end{gathered} \]
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1In 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.
