The Clowder Project

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*}
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  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

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