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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2.
A category $\mathcal{C}$ is posetal1 if $\mathcal{C}$ is equivalent to $X_{\mathsf{pos}}$ for some poset $\webleft (X,\preceq _{X}\webright )$.
- 1Further Terminology: Also called a thin category or a $\webleft (0,1\webright )$-category.
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Functoriality. The assignment $\webleft (X,\preceq _{X}\webright )\mapsto X_{\mathsf{pos}}$ defines a functor
\[ \webleft (-\webright )_{\mathsf{pos}}\colon \mathsf{Pos}\to \mathsf{Cats}. \] -
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Fully Faithfulness. The functor $\webleft (-\webright )_{\mathsf{pos}}$ of Item 1 is fully faithful.
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Characterisations. The following conditions are equivalent:
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Automatic Commutativity of Diagrams. Every diagram in a posetal category commutes.
Let $\webleft (X,\preceq _{X}\webright )$ be a poset and let $\mathcal{C}$ be a category.
Proof of Proposition 11.2.7.1.2.
Item 1: Functoriality
Omitted.
Item 2: Fully Faithfulness
Omitted.
Item 3: Characterisations
Clear.
Item 4: Automatic Commutativity of Diagrams
This follows from the fact that if $\mathcal{C}$ is posetal, then there’s at most one morphism between any two objects.
