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

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

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

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

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

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

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

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

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

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

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

     \[ ([B]\to [C])\circ ([A]\to [B]) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}([A]\to [C]). \]
2. 2.

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