Kerodon

Example 10.2.6.18. Let $\operatorname{Ret}$ be the category introduced in Construction 8.5.0.2: that is, the category which is freely generated by a pair of morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ satisfying the identity $r \circ i= \operatorname{id}_{Y}$. By virtue of Exercise 8.5.0.3, there is a unique functor $\operatorname{Ret}\rightarrow \operatorname{{\bf \Delta }}$ which carries $i$ to the inclusion map $[0] \hookrightarrow [1]$ and $r$ to the constant function $[1] \twoheadrightarrow [0]$. This functor induces an isomorphism from $\operatorname{Ret}$ onto the subcategory $\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq 1} \subset \operatorname{{\bf \Delta }}$ of Notation 10.2.6.16.