Let $\webleft (X,\preceq _{X}\webright )$ be a poset.
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The posetal category associated to $\webleft (X,\preceq _{X}\webright )$ is the category $X_{\mathsf{pos}}$ where
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Objects. We have
\[ \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{pos}}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \] -
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Morphisms. For each $a,b\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{pos}}\webright )$, we have
\[ \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathrm{pt}& \text{if $a\preceq _{X}b$},\\ \text{Ø}& \text{otherwise.}\end{cases} \] -
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Identities. For each $a\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{pos}}\webright )$, the unit map
\[ \mathbb {1}^{X_{\mathsf{pos}}}_{a}\colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}\webleft (a,a\webright ) \]of $X_{\mathsf{pos}}$ at $a$ is given by the identity map.
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Composition. For each $a,b,c\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{pos}}\webright )$, the composition map
\[ \circ ^{X_{\mathsf{pos}}}_{a,b,c}\colon \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}\webleft (b,c\webright )\times \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}\webleft (a,b\webright )\to \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}\webleft (a,c\webright ) \]of $X_{\mathsf{pos}}$ at $\webleft (a,b,c\webright )$ is defined as either the inclusion $\text{Ø}\hookrightarrow \mathrm{pt}$ or the identity map of $\mathrm{pt}$, depending on whether we have $a\preceq _{X}b$, $b\preceq _{X}c$, and $a\preceq _{X}c$.
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