The Clowder Project

The punctual category¹ is the category $\mathsf{pt}$ where

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  Objects. We have

  \[ \operatorname {\mathrm{Obj}}\webleft (\mathsf{pt}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \star \right\} . \]
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  Morphisms. The unique $\operatorname {\mathrm{Hom}}$-set of $\mathsf{pt}$ is defined by

  \[ \operatorname {\mathrm{Hom}}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \operatorname {\mathrm{id}}_{\star }\right\} . \]
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  Identities. The unit map

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

  of $\mathsf{pt}$ at $\star $ is defined by

  \[ \operatorname {\mathrm{id}}^{\mathsf{pt}}_{\star } \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{\star }. \]
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  Composition. The composition map

  \[ \circ ^{\mathsf{pt}}_{\star ,\star ,\star } \colon \operatorname {\mathrm{Hom}}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \times \operatorname {\mathrm{Hom}}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \to \operatorname {\mathrm{Hom}}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \]

  of $\mathsf{pt}$ at $\webleft (\star ,\star ,\star \webright )$ is given by the bijection $\mathrm{pt}\times \mathrm{pt}\cong \mathrm{pt}$.