Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 8.5.0.1. We define a category $\operatorname{Ret}$ as follows:

  • The category $\operatorname{Ret}$ has exactly two objects, which we denote by $\widetilde{X}$ and $\widetilde{Y}$.

  • The morphisms sets in $\operatorname{Ret}$ are given by

    \[ \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{X}, \widetilde{X}) = \{ \operatorname{id}_{\widetilde{X}}, \widetilde{e} \} \quad \quad \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{X}, \widetilde{Y}) = \{ \widetilde{r} \} \]
    \[ \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{Y}, \widetilde{X}) = \{ \widetilde{i} \} \quad \quad \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{Y}, \widetilde{Y}) = \{ \operatorname{id}_{\widetilde{Y}} \} . \]
  • The composition law in $\operatorname{Ret}$ is given (on non-identity morphisms) by the formulae

    \[ \widetilde{r} \circ \widetilde{i} = \operatorname{id}_{\widetilde{Y}} \quad \quad \widetilde{i} \circ \widetilde{r} = \widetilde{e} \]
    \[ \widetilde{e} \circ \widetilde{i} = \widetilde{i} \quad \quad \widetilde{e} \circ \widetilde{e} = \widetilde{e} \quad \quad \widetilde{r} \circ \widetilde{e} = \widetilde{r}. \]