The Clowder Project

Let $\webleft (X,\preceq _{X}\webright )$ be a poset.

1. The posetal category associated to $\webleft (X,\preceq _{X}\webright )$ is the category $X_{\mathsf{pos}}$ where
   • Objects. We have
     \[ \text{Obj}\webleft (X_{\mathsf{pos}}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \]

   • Morphisms. For each $a,b\in \text{Obj}\webleft (X_{\mathsf{pos}}\webright )$, we have
     \[ \textup{Hom}_{X_{\mathsf{pos}}}\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \text{pt}& \text{if $a\preceq _{X}b$},\\ \text{Ø}& \text{otherwise.}\end{cases} \]

   • Identities. For each $a\in \text{Obj}\webleft (X_{\mathsf{pos}}\webright )$, the unit map
     \[ \mathbb {1}^{X_{\mathsf{pos}}}_{a}\colon \text{pt}\to \textup{Hom}_{X_{\mathsf{pos}}}\webleft (a,a\webright ) \]

     of $X_{\mathsf{pos}}$ at $a$ is given by the identity map.

   • Composition. For each $a,b,c\in \text{Obj}\webleft (X_{\mathsf{pos}}\webright )$, the composition map
     \[ \circ ^{X_{\mathsf{pos}}}_{a,b,c}\colon \textup{Hom}_{X_{\mathsf{pos}}}\webleft (b,c\webright )\times \textup{Hom}_{X_{\mathsf{pos}}}\webleft (a,b\webright )\to \textup{Hom}_{X_{\mathsf{pos}}}\webleft (a,c\webright ) \]

     of $X_{\mathsf{pos}}$ at $\webleft (a,b,c\webright )$ is defined as either the inclusion $\text{Ø}\hookrightarrow \text{pt}$ or the identity map of $\text{pt}$, depending on whether we have $a\preceq _{X}b$, $b\preceq _{X}c$, and $a\preceq _{X}c$.

2. A category $\mathcal{C}$ is posetal¹ if $\mathcal{C}$ is equivalent to $X_{\mathsf{pos}}$ for some poset $\webleft (X,\preceq _{X}\webright )$.