Item 2 (00YQ) — The Clowder Project

11.3.3 Discrete Categories

Definition 11.3.3.1.1 ▶ Discrete Categories

Let $X$ be a set.

1.
The discrete category on $X$ is the category $X_{\mathsf{disc}}$ where
- •
  Objects. We have
  
  \[ \operatorname {\mathrm{Obj}}(X_{\mathsf{disc}}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \]
- •
  Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(X_{\mathsf{disc}})$, we have
  
  \[ \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}(A,B) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \operatorname {\mathrm{id}}_{A} & \text{if $A=B$,}\\ \text{Ø}& \text{if $A\neq B$.} \end{cases} \]
- •
  Identities. For each $A\in \operatorname {\mathrm{Obj}}(X_{\mathsf{disc}})$, the unit map
  
  \[ \mathbb {1}^{X_{\mathsf{disc}}}_{A} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}(A,A) \]
  
  of $X_{\mathsf{disc}}$ at $A$ is defined by
  
  \[ \operatorname {\mathrm{id}}^{X_{\mathsf{disc}}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{A}. \]
- •
  Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}(X_{\mathsf{disc}})$, the composition map
  
  \[ \circ ^{X_{\mathsf{disc}}}_{A,B,C} \colon \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}(B,C) \times \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}(A,B) \to \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}(A,C) \]
  
  of $X_{\mathsf{disc}}$ at $(A,B,C)$ is defined by
  
  \[ \operatorname {\mathrm{id}}_{A}\circ \operatorname {\mathrm{id}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{A}. \]
2.
A category $\mathcal{C}$ is discrete if it is equivalent to $X_{\mathsf{disc}}$ for some set $X$.

Proposition 11.3.3.1.2 ▶ Properties of Discrete Categories on Sets

Let $X$ be a set.

1.
Functoriality. The assignment $X\mapsto X_{\mathsf{disc}}$ defines a functor

\[ (-)_{\mathsf{disc}} \colon \mathsf{Sets}\to \mathsf{Cats} \]

where:
- •
  Action on Objects. For each $X\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, we have
  
  \[ [(-)_{\mathsf{disc}}](X)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X_{\mathsf{disc}} \]
- •
  Action on Morphisms. Given a function $f\colon X\to Y$, the image
  
  \[ f_{\mathsf{disc}}\colon X_{\mathsf{disc}}\to Y_{\mathsf{disc}} \]
  
  of $f$ by $(-)_{\mathsf{disc}}$ is the functor given by $x\mapsto f(x)$ on objects and $\operatorname {\mathrm{id}}_{x}\mapsto \operatorname {\mathrm{id}}_{f(x)}$ on morphisms.

2.
Adjointness. We have a quadruple adjunction

Symmetric Strong Monoidality With Respect to Coproducts. The functor of Item 1 has a symmetric strong monoidal structure

\[ ((-)_{\mathsf{disc}},(-)^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}},(-)^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}}) \colon (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}) \to (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}_{\mathsf{cat}}), \]

being equipped with isomorphisms

\[ \begin{gathered} (-)^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y_{\mathsf{disc}} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }(X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y)_{\mathsf{disc}},\\ (-)^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}} \colon \text{Ø}_{\mathsf{cat}}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{Ø}_{\mathsf{disc}}, \end{gathered} \]

natural in $X,Y\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$.

Symmetric Strong Monoidality With Respect to Products. The functor of Item 1 has a symmetric strong monoidal structure

\[ ((-)_{\mathsf{disc}},(-)^{\times }_{\mathsf{disc}},(-)^{\times }_{\mathsf{disc}|\mathbb {1}}) \colon (\mathsf{Sets},\times ,\mathrm{pt}) \to (\mathsf{Cats},\times ,\mathsf{pt}), \]

being equipped with isomorphisms

\[ \begin{gathered} (-)^{\times }_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\times Y_{\mathsf{disc}} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }(X\times Y)_{\mathsf{disc}},\\ (-)^{\times }_{\mathsf{disc}|\mathbb {1}} \colon \mathsf{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{pt}_{\mathsf{disc}}, \end{gathered} \]

natural in $X,Y\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$.

Proof of Proposition 11.3.3.1.2.

Item 1: Functoriality

First, note that since the Hom-sets of $Y_{\mathsf{disc}}$ all have at most one element, it follows that the preservation of identities and composition conditions for functors hold automatically for $f_{\mathsf{disc}}$, so it is indeed a functor.

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

•
Preservation of Identities. For each $X\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, we have

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

for each $x\in X$. Since there’s at most one element in the Hom-sets of $X_{\mathsf{disc}}$, the corresponding equalities also hold for morphisms. Thus $(\operatorname {\mathrm{id}}_{X})_{\mathsf{disc}}=\operatorname {\mathrm{id}}_{X_{\mathsf{disc}}}$.
•
Preservation of Composition. Given functions $f\colon X\to Y$ and $g\colon Y\to Z$, we have

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

for each $x\in X$. Again, the equations must also hold on morphisms due to there being at most one element in the Hom-sets of $Z_{\mathsf{disc}}$. Thus $(g\circ f)_{\mathsf{disc}}=g_{\mathsf{disc}}\circ f_{\mathsf{disc}}$.