Item 3a (00XA) — The Clowder Project

11.2.7 Posetal Categories

Definition 11.2.7.1.1 ▶ Posetal Categories

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

1.
The posetal category associated to $(X,\preceq _{X})$ is the category $X_{\mathsf{pos}}$ where
- •
  Objects. We have
  
  \[ \operatorname {\mathrm{Obj}}(X_{\mathsf{pos}})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \]
- •
  Morphisms. For each $a,b\in \operatorname {\mathrm{Obj}}(X_{\mathsf{pos}})$, we have
  
  \[ \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(a,b)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathrm{pt}& \text{if $a\preceq _{X}b$},\\ \text{Ø}& \text{otherwise.}\end{cases} \]
- •
  Identities. For each $a\in \operatorname {\mathrm{Obj}}(X_{\mathsf{pos}})$, the unit map
  
  \[ \mathbb {1}^{X_{\mathsf{pos}}}_{a}\colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(a,a) \]
  
  of $X_{\mathsf{pos}}$ at $a$ is given by the identity map.
- •
  Composition. For each $a,b,c\in \operatorname {\mathrm{Obj}}(X_{\mathsf{pos}})$, the composition map
  
  \[ \circ ^{X_{\mathsf{pos}}}_{a,b,c}\colon \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(b,c)\times \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(a,b)\to \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(a,c) \]
  
  of $X_{\mathsf{pos}}$ at $(a,b,c)$ 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 $a\preceq _{X}c$.
2.
A category $\mathcal{C}$ is posetal¹ if $\mathcal{C}$ is equivalent to $X_{\mathsf{pos}}$ for some poset $(X,\preceq _{X})$.

¹Further Terminology: Also called a thin category or a $(0,1)$-category.

Proposition 11.2.7.1.2 ▶ Properties of Posetal Categories

Let $(X,\preceq _{X})$ be a poset and let $\mathcal{C}$ be a category.

1.
Functoriality. The assignment $(X,\preceq _{X})\mapsto X_{\mathsf{pos}}$ defines a functor

\[ (-)_{\mathsf{pos}}\colon \mathsf{Pos}\to \mathsf{Cats}. \]

where:
- •
  Action on Objects. For each $X\in \operatorname {\mathrm{Obj}}(\mathsf{Pos})$, we have
  
  \[ [(-)_{\mathsf{pos}}](X)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X_{\mathsf{pos}}, \]
  
  where $X_{pos}$ is the category of of Item 1 of Definition 11.2.7.1.1.
- •
  Action on Morphisms. Given a morphism of posets $f\colon X\to Y$ in $\mathsf{Pos}$, the image
  
  \[ f_{\mathsf{pos}}\colon X_{\mathsf{pos}}\to Y_{\mathsf{pos}} \]
  
  of $f$ by $(-)_{\mathsf{pos}}$ is the functor defined as follows:
  - •
    The Action of $f_{\mathsf{pos}}$ on Objects. For each $x\in \operatorname {\mathrm{Obj}}(X_{\mathsf{pos}})$, we have
    
    \[ f_{\mathsf{pos}}(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f(x). \]
  - •
    The Action of $f_{\mathsf{pos}}$ on Morphisms. For each $x,y\in \operatorname {\mathrm{Obj}}(X_{\mathsf{pos}})$, the action
    
    \[ f_{\mathsf{pos}|x,y}\colon \operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(x,y)\to \operatorname {\mathrm{Hom}}_{Y_{\mathsf{pos}}}(f(x),f(y)) \]
    
    of $f$ at $(x,y)$ is given by
    
    \[ f_{\mathsf{pos}|x,y}(\mathrm{pt}_{\operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(x,y)})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{pt}_{\operatorname {\mathrm{Hom}}_{Y_{\mathsf{pos}}}(f(x),f(y))} \]
    
    if $x\preceq _{X}y$ or, otherwise, by the inclusion of the empty set into $\operatorname {\mathrm{Hom}}_{Y_{\mathsf{pos}}}(f(x),f(y))$.
2.
Fully Faithfulness. The functor $(-)_{\mathsf{pos}}$ of Item 1 is fully faithful.
3.
Characterisations. The following conditions are equivalent:

(a)
The category $\mathcal{C}$ is posetal.

(b)

For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$ and each $f,g\in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$, we have $f=g$.

Automatic Commutativity of Diagrams. Every diagram in a posetal category commutes.

Proof of Proposition 11.2.7.1.2.

Item 1: Functoriality

First, note that given a morphism of posets $f\colon X\to Y$, the corresponding functor $f_{\mathsf{pos}}\colon X_{\mathsf{pos}}\to Y_{\mathsf{pos}}$ is indeed a functor: since all morphisms in the Hom-sets of $Y_{\mathsf{pos}}$ are equal, it preserves identities and compositions trivially.

Next, we claim that $(-)_{\mathsf{pos}}$ is indeed a functor:

•
Preservation of Identities. Let $X\in \operatorname {\mathrm{Obj}}(\mathsf{Pos})$. Given $x,y\in X$ with $x\preceq _{X}y$, we have

\begin{align*} (\operatorname {\mathrm{id}}_{X})_{\mathsf{pos}}(x) & = \operatorname {\mathrm{id}}_{X}(x)\\ & = \operatorname {\mathrm{id}}_{X_{\mathsf{pos}}}(x), \end{align*}

so $(\operatorname {\mathrm{id}}_{X})_{\mathsf{pos}}$ acts like the identity functor of $X_{\mathsf{pos}}$ on objects, and

\begin{align*} (\operatorname {\mathrm{id}}_{X})_{\mathsf{pos}}(\mathrm{pt}_{\operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(x,y)}) & = \mathrm{pt}_{\operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}((\operatorname {\mathrm{id}}_{X})_{\mathsf{pos}}(x),(\operatorname {\mathrm{id}}_{X})_{\mathsf{pos}}(y))}\\ & = \mathrm{pt}_{\operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(a,b)}, \end{align*}

so the same holds for morphisms. Thus $(\operatorname {\mathrm{id}}_{X})_{\mathsf{pos}}=\operatorname {\mathrm{id}}_{X_{\mathsf{pos}}}$.
•
Preservation of Composition. Let $X,Y,Z\in \operatorname {\mathrm{Obj}}(\mathsf{Pos})$. Given morphisms of posets $f\colon X\to Y$ and $g\colon Y\to Z$, we need to show

\[ (g\circ f)_{\mathsf{pos}}=g_{\mathsf{pos}}\circ f_{\mathsf{pos}}. \]

Indeed, given $x\in X$, we have

\begin{align*} (g\circ f)_{\mathsf{pos}}(x) & = (g\circ f)(x)\\ & = g(f(x))\\ & = g_{\mathsf{pos}}(f_{\mathsf{pos}}(x))\\ & = [g_{\mathsf{pos}}\circ f_{\mathsf{pos}}](x), \end{align*}

so the identity holds on objects. Since $Z_{\mathsf{pos}}$ is a posetal category, the identity automatically holds on morphisms since

\begin{align*} (g\circ f)_{\mathsf{pos}}(\mathrm{pt}_{\operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(x,y)}) & = \mathrm{pt}_{\operatorname {\mathrm{Hom}}_{Z_{\mathsf{pos}}}(g_{\mathsf{pos}}(f_{\mathsf{pos}}(x)),g_{\mathsf{pos}}(f_{\mathsf{pos}}(y)))}\\ & = [g_{\mathsf{pos}}\circ f_{\mathsf{pos}}](\mathrm{pt}_{\operatorname {\mathrm{Hom}}_{X_{\mathsf{pos}}}(x,y)}) \end{align*}

for each $x,y\in X$ with $x\preceq _{X}y$.