Proposition 1.3.2.7 (04G2)

Proposition 1.3.2.7. The construction $M \mapsto B_{\bullet }M$ determines a fully faithful embedding $\operatorname{Mon}\hookrightarrow \operatorname{Set_{\Delta }}$. The essential image of this functor consists of those simplicial sets $S_{\bullet }$ which satisfy the following condition for each $n \geq 0$:

$(\ast _ n)$: For $1 \leq i \leq n$, let $\rho _ i: S_{n} \rightarrow S_{1}$ denote the map associated to the inclusion of linearly ordered sets $[1] \simeq \{ i-1, i \} \hookrightarrow [n]$. Then the maps $\{ \rho _ i \} _{1 \leq i \leq n}$ determine a bijection $S_{n} \rightarrow \prod _{1 \leq i \leq n} S_1$.

Proof of Proposition 1.3.2.7. We first show that Construction 1.3.2.5 is fully faithful. Fix monoids $M$ and $M'$ and let $f_{\bullet }: B_{\bullet } M \rightarrow B_{\bullet } M'$ be a morphism of simplicial sets. Applying Proposition 1.3.2.14 (together with Remark 1.3.2.13), we deduce that there is a unique nonunital monoid homomorphism $g: M \rightarrow M'$ such that $f_{\bullet }$ coincides with $B_{\bullet } g$ (as a morphism of semisimplicial sets). Since $f_{\bullet }$ is a morphism of simplicial sets, it carries the (unique) degenerate edge of $B_{\bullet } M$ to the (unique) degenerate edge of $B_{\bullet } M'$. It follows that $g$ carries the unit element of $M$ to the unit element of $M'$: that is, it is a monoid homomorphism.

Now suppose that $S_{\bullet }$ is a simplicial set satisfying condition $(\ast _ n)$ for each $n \geq 0$. Applying Proposition 1.3.2.14, we deduce that there is a nonunital monoid $M$ and an isomorphism of semisimplicial sets $f_{\bullet }: B_{\bullet } M \rightarrow S_{\bullet }$, which carries each $n$-tuple $(x_ n, \cdots , x_1) \in M$ to the $n$-simplex $\sigma _{ x_ n, \cdots , x_1}$ of $S_{\bullet }$ appearing in the proof of Proposition 1.3.2.14. Let $e \in M$ be the element corresponding to the unique degenerate $1$-simplex of $S_{\bullet }$. For $0 \leq i \leq n$, the degeneracy operator $s^{n}_ i: S_{n} \rightarrow S_{n+1}$ satisfies the identity

1.29

\begin{equation} \label{equation:monoids-as-semisimplicial-objects} s^{n}_ i( \sigma _{x_{n}, \cdots , x_1} ) = \sigma _{ x_ n, \cdots , x_{i+1}, e, x_ i, \cdots , x_1}. \end{equation}

Specializing to the case $i=n=1$ and applying the face operator $d^{1}_1$, we obtain an equality

\begin{eqnarray*} \sigma _{x} & = & d^{2}_1( s^{1}_1( \sigma _{x} ) ) \\ & = & d^{2}_1( \sigma _{e,x} ) \\ & = & \sigma _{ex}; \end{eqnarray*}

that is, $e$ is a left unit with respect to the multiplication on $M$. A similar argument shows that $e$ is a right unit with respect to the multiplication on $M$: that is, $M$ is a monoid with unit element $e$. To complete the proof, it will suffice to show that $f_{\bullet }: B_{\bullet } M \rightarrow S_{\bullet }$ is an isomorphism of simplicial sets: that is, it commutes with degeneracy operators as well as face operators. This is a restatement of the identity (1.29). $\square$