Kerodon

Proof. We proceed as in the proof of Proposition 1.1.1.9. Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ and a collection of morphisms $\{ \widetilde{\sigma }_{n}^{i}: [n+1] \rightarrow [n] \} _{0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ obtained by imposing the relation

for every triple of integers $0 \leq i \leq j \leq n$. Using Remark 1.1.2.11, we see that there is a unique functor $F_{\operatorname{surj}}: \overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}} \rightarrow \operatorname{{\bf \Delta }}_{\operatorname{surj}}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ to itself, and each generating morphism $\widetilde{\sigma }_{n}^{i}$ to the epimorphism $\sigma _{n}^{i}: [n+1] \twoheadrightarrow [n]$ of Construction 1.1.2.1. To prove Proposition 1.1.2.13, it will suffice to show that the functor $F_{\operatorname{surj}}$ is an isomorphism of categories.

Fix integers $0 \leq m \leq n$, and set $b = n-m+1$. In the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$, every morphism $\beta : [n] \rightarrow [m]$ admits a unique factorization $\beta = \widetilde{\sigma }_{m}^{i_0} \circ \widetilde{\sigma }_{m+1}^{i_1} \circ \cdots \circ \widetilde{\sigma }_{m+b}^{i_{b}}$, where the superscripts are nonnegative integers satisfying $0 \leq i_ a \leq m+a$ for $0 \leq a \leq b$. Let us say that $\beta $ is in standard form if, in addition, the integers $i_ a$ satisfy the inequalities $i_0 < i_1 < i_2 < \cdots < i_ b$. Note that, by repeatedly applying the relation (1.10), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ to a morphism which is in standard form. More precisely, every morphism $\overline{\beta }: [n] \rightarrow [m]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ which is in standard form.

By construction, the functor $F_{\operatorname{surj}}$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [n] \twoheadrightarrow [m]$ in $\operatorname{{\bf \Delta }}_{\operatorname{surj}}$, there is a unique morphism $\overline{\beta }: [n] \rightarrow [m]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ satisfying $F_{\operatorname{surj}}( \overline{\beta } ) = \alpha $. By virtue of the preceding discussion, it will suffice to show that $\alpha $ can be lifted uniquely to a morphism $\beta : [n] \rightarrow [m]$ in the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ which is in standard form. We now observe that $\beta =\widetilde{\sigma }_{m}^{i_0} \circ \widetilde{\sigma }_{m+1}^{i_1} \circ \cdots \circ \widetilde{\sigma }_{m+b}^{i_{b}}$ is characterized by the requirement that $\{ i_0 < i_1 < \cdots < i_ b \} $ is the collection of integers $0 \leq j < n$ satisfying $\alpha (j) = \alpha (j+1)$. $\square$