The Clowder Project

Let $X$ be a set.

1. The indiscrete category on $X$^[1] is the category $X_{\mathsf{indisc}}$ where
   • Objects. We have
     \[ \text{Obj}\webleft (X_{\mathsf{indisc}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \]

   • Morphisms. For each $A,B\in \text{Obj}\webleft (X_{\mathsf{indisc}}\webright )$, we have
     \begin{align*} \textup{Hom}_{X_{\mathsf{disc}}}\webleft (A,B\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft [A\webright ]\to \webleft [B\webright ]\webright\} \\ & \cong \text{pt}. \end{align*}

   • Identities. For each $A\in \text{Obj}\webleft (X_{\mathsf{indisc}}\webright )$, the unit map
     \[ \mathbb {1}^{X_{\mathsf{indisc}}}_{A} \colon \text{pt}\to \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (A,A\webright ) \]

     of $X_{\mathsf{indisc}}$ at $A$ is defined by

     \[ \text{id}^{X_{\mathsf{indisc}}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft [A\webright ]\to \webleft [A\webright ]\webright\} . \]

   • Composition. For each $A,B,C\in \text{Obj}\webleft (X_{\mathsf{indisc}}\webright )$, the composition map
     \[ \circ ^{X_{\mathsf{indisc}}}_{A,B,C} \colon \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (B,C\webright ) \times \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (A,B\webright ) \to \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (A,C\webright ) \]

     of $X_{\mathsf{disc}}$ at $\webleft (A,B,C\webright )$ is defined by

     \[ \webleft (\webleft [B\webright ]\to \webleft [C\webright ]\webright )\circ \webleft (\webleft [A\webright ]\to \webleft [B\webright ]\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\webleft [A\webright ]\to \webleft [C\webright ]\webright ). \]

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