The $n$th ordinal category is the category $\mathbb {n}$ where1
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Objects. We have
\[ \operatorname {\mathrm{Obj}}(\mathbb {n}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ [0],\ldots ,[n]\right\} . \] -
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Morphisms. For each $[i],[j]\in \operatorname {\mathrm{Obj}}(\mathbb {n})$, we have
\[ \operatorname {\mathrm{Hom}}_{\mathbb {n}}([i],[j]) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \left\{ \operatorname {\mathrm{id}}_{[i]}\right\} & \text{if $[i]=[j]$,}\\ \left\{ [i]\to [j]\right\} & \text{if $[j]<[i]$,}\\ \text{Ø}& \text{if $[j]>[i]$.} \end{cases} \] -
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Identities. For each $[i]\in \operatorname {\mathrm{Obj}}(\mathbb {n})$, the unit map
\[ \mathbb {1}^{\mathbb {n}}_{[i]} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{\mathbb {n}}([i],[i]) \]of $\mathbb {n}$ at $[i]$ is defined by
\[ \operatorname {\mathrm{id}}^{\mathbb {n}}_{[i]} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{[i]}. \] -
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Composition. For each $[i],[j],[k]\in \operatorname {\mathrm{Obj}}(\mathbb {n})$, the composition map
\[ \circ ^{\mathbb {n}}_{[i],[j],[k]} \colon \operatorname {\mathrm{Hom}}_{\mathbb {n}}([j],[k]) \times \operatorname {\mathrm{Hom}}_{\mathbb {n}}([i],[j]) \to \operatorname {\mathrm{Hom}}_{\mathbb {n}}([i],[k]) \]of $\mathbb {n}$ at $([i],[j],[k])$ is defined by
\[ \begin{gathered} \operatorname {\mathrm{id}}_{[i]}\circ \operatorname {\mathrm{id}}_{[i]} = \operatorname {\mathrm{id}}_{[i]},\\ ([j]\to [k])\circ ([i]\to [j]) = ([i]\to [k]).\end{gathered} \]
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1In other words, $\mathbb {n}$ is the category associated to the poset \[ [0]\to [1]\to \cdots \to [n-1]\to [n]. \]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 (\mathbb {0}\star \mathbb {0})\star \mathbb {0},\\ \mathbb {3} & \cong \mathbb {2}\star \mathbb {0}\\ & \cong (\mathbb {1}\star \mathbb {0})\star \mathbb {0}\\ & \cong ((\mathbb {0}\star \mathbb {0})\star \mathbb {0})\star \mathbb {0},\\ \mathbb {4} & \cong \mathbb {3}\star \mathbb {0}\\ & \cong (\mathbb {2}\star \mathbb {0})\star \mathbb {0}\\ & \cong ((\mathbb {1}\star \mathbb {0})\star \mathbb {0})\star \mathbb {0}\\ & \cong (((\mathbb {0}\star \mathbb {0})\star \mathbb {0})\star \mathbb {0})\star \mathbb {0}, \end{align*}and so on.
