The Clowder Project

Let $X$ be a set.

1. 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
     \[ \textup{Hom}_{X_{\mathsf{disc}}}\webleft (A,B\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \text{id}_{A} & \text{if $A=B$,}\\ \text{Ø}& \text{if $A\neq B$.} \end{cases} \]

   • 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

     \[ \text{id}_{A}\circ \text{id}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{A}. \]

2. A category $\mathcal{C}$ is discrete if it is equivalent to $X_{\mathsf{disc}}$ for some set $X$.