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

Proposition Let $\operatorname{\mathcal{C}}$ be a category containing a sequence of objects $\{ C_{n} \} _{n \geq 0}$. Then morphisms

\[ \{ d^{n}_{i}: C_{n} \rightarrow C_{n-1} \} _{ 0 \leq i \leq n, n > 0} \quad \quad \{ s^{n}_{i}: C_{n} \rightarrow C_{n+1} \} _{0 \leq i \leq n} \]

are the face and degeneracy operators for a simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast )$ of Remark, condition $(\ast ')$ of Exercise, and condition $(\ast '')$ of Remark, and

Proof. We proceed as in the proofs of Propositions and Let $\widetilde{\operatorname{{\bf \Delta }}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ together with morphisms $\{ \widetilde{\delta }_{n}^{i}: [n-1] \rightarrow [n] \} _{n > 0, 0 \leq i \leq n}$ and $\{ \widetilde{\sigma }_{n}^{i}: [n+1] \rightarrow [n] \} _{0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}$ obtained by imposing the relations (1.4) and (1.10), together with the following:

\begin{eqnarray} \label{equation:relation-semisimplicial-identity3} \widetilde{\sigma }^{j}_{n} \circ \widetilde{\delta }_{n+1}^{i} & = & \begin{cases} \widetilde{\delta }_{n}^{i} \circ \widetilde{\sigma }^{j-1}_{n-1} & \text{ if } i < j \\ \operatorname{id}_{ [n] } & \text{ if } i = j \text{ or } i = j + 1 \\ \widetilde{\delta }^{i-1}_{n} \circ \widetilde{\sigma }^{j}_{n-1} & \text{ if } i > j+1. \end{cases}\end{eqnarray}

for every triple of integers $0 \leq i , j \leq n$. There is a unique functor $F: \overline{\operatorname{{\bf \Delta }}} \rightarrow \operatorname{{\bf \Delta }}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}$ to itself and satisfies $F( \widetilde{\delta }_{n}^{i} ) = \delta _{n}^{i}$ and $F( \widetilde{\sigma }_{n}^{i} ) = \sigma _{n}^{i}$. To prove Proposition, it will suffice to show that the functor $F$ is an isomorphism of categories.

Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ be the categories appearing in the proofs of Proposition and Proposition, respectively. Let us identify $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ with (non-full) subcategories of $\widetilde{\operatorname{{\bf \Delta }}}$. We will say that a morphism $\beta : [m] \rightarrow [n]$ of $\widetilde{\operatorname{{\bf \Delta }}}$ is weakly standard if it factors as a composition $[m] \xrightarrow { \beta _{\operatorname{surj}} } [k] \xrightarrow { \beta _{\operatorname{inj}} } [n]$, where $\beta _{\operatorname{inj}}$ belongs to $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ belongs to $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$. In this case, the morphisms $\beta _{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ are uniquely determined. We will say that $\beta $ is in standard form if it is weakly standard and, in addition, the morphisms $\beta _{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ are in standard form (as in the proofs of Propositions and Note that, by repeatedly applying the relation (1.11), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}$ into a morphism $\beta $ which is weakly standard. Using the relations (1.4) and (1.10), we can further arrange that $\beta $ is in standard form. It follows that every morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ of $\widetilde{\operatorname{{\bf \Delta }}}$ which is in standard form.

By construction, the functor $F$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, there is a unique morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}$ satisfying $F( \overline{\beta } ) = \alpha $. Let $\widetilde{F}$ denote the composite functor $\widetilde{\operatorname{{\bf \Delta }}} \twoheadrightarrow \overline{\operatorname{{\bf \Delta }}} \xrightarrow {F} \operatorname{{\bf \Delta }}$. By virtue of the preceding discussion, it will suffice to show that there is a unique morphism $\beta : [m] \rightarrow [n]$ in $\widetilde{ \operatorname{{\bf \Delta }}}$ which is in standard form and satisfies $\widetilde{F}( \beta ) = \alpha $. In the simplex category $\operatorname{{\bf \Delta }}$, the morphism $\alpha $ factors uniquely as a composition $[m] \xrightarrow { \alpha _{\operatorname{surj}} } [k] \xrightarrow { \alpha _{\operatorname{inj}} } [n]$, where $\alpha _{\operatorname{inj}}$ is an injection and $\alpha _{\operatorname{surj}}$ is a surjection. If $\beta : [m] \rightarrow [n]$ is a weakly standard morphism of $\widetilde{\operatorname{{\bf \Delta }}}$, then the identity $\widetilde{F}( \beta ) = \alpha $ holds if and only if $\widetilde{F}( \beta _{\operatorname{inj}} ) = \alpha _{\operatorname{inj}}$ and $\widetilde{F}( \beta _{\operatorname{surj}} ) = \alpha _{\operatorname{surj}}$. We are therefore reduced to proving that $\alpha _{\operatorname{inj}}$ and $\alpha _{\operatorname{surj}}$ can be lifted uniquely to morphisms of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ which are in standard form, which was established in the proofs of Proposition and Proposition $\square$