The Clowder Project

Let $X$ be a set.

1. 1.

   The indiscrete category on $X$¹ is the category $X_{\mathsf{indisc}}$ where

   • •

     Objects. We have

     \[ \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{indisc}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \]
   • •

     Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{indisc}}\webright )$, we have

     \begin{align*} \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}\webleft (A,B\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \webleft [A\webright ]\to \webleft [B\webright ]\right\} \\ & \cong \mathrm{pt}. \end{align*}
   • •

     Identities. For each $A\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{indisc}}\webright )$, the unit map

     \[ \mathbb {1}^{X_{\mathsf{indisc}}}_{A} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{X_{\mathsf{indisc}}}\webleft (A,A\webright ) \]

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

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

     Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{indisc}}\webright )$, the composition map

     \[ \circ ^{X_{\mathsf{indisc}}}_{A,B,C} \colon \operatorname {\mathrm{Hom}}_{X_{\mathsf{indisc}}}\webleft (B,C\webright ) \times \operatorname {\mathrm{Hom}}_{X_{\mathsf{indisc}}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Hom}}_{X_{\mathsf{indisc}}}\webleft (A,C\webright ) \]

     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. 2.

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