Subsection 8.2.3 (00YJ): Discrete Categories

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Let $X$ be a set.

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

Let $X$ be a set.

Functoriality. The assignment $X\mapsto X_{\mathsf{disc}}$ defines a functor
\[ \webleft (-\webright )_{\mathsf{disc}} \colon \mathsf{Sets}\to \mathsf{Cats}. \]
Adjointness. We have a quadruple adjunction
Symmetric Strong Monoidality With Respect to Coproducts. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )_{\mathsf{disc}},\webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}},\webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset \webright ) \to \webleft (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset _{\mathsf{cat}}\webright ), \]

being equipped with isomorphisms

\[ \begin{gathered} \webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y_{\mathsf{disc}} \xrightarrow {\cong }\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )_{\mathsf{disc}},\\ \webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}} \colon \emptyset _{\mathsf{cat}}\xrightarrow {\cong }\emptyset _{\mathsf{disc}}, \end{gathered} \]

natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Symmetric Strong Monoidality With Respect to Products. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )_{\mathsf{disc}},\webleft (-\webright )^{\times }_{\mathsf{disc}},\webleft (-\webright )^{\times }_{\mathsf{disc}|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \to \webleft (\mathsf{Cats},\times ,\mathsf{pt}\webright ), \]

being equipped with isomorphisms

\[ \begin{gathered} \webleft (-\webright )^{\times }_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\times Y_{\mathsf{disc}} \xrightarrow {\cong }\webleft (X\times Y\webright )_{\mathsf{disc}},\\ \webleft (-\webright )^{\times }_{\mathsf{disc}|\mathbb {1}} \colon \mathsf{pt}\xrightarrow {\cong }\text{pt}_{\mathsf{disc}}, \end{gathered} \]

natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.

Item 1: Functoriality

Clear.

Item 2: Adjointness

This is proved in Proposition 8.2.1.1.1.

Item 3: Symmetric Strong Monoidality With Respect to Coproducts

Clear.

Item 4: Symmetric Strong Monoidality With Respect to Products

Clear.

The Clowder Project

8.2.3 Discrete Categories