Kerodon

Proof. We first show that the functor $M \mapsto B_{\bullet } M$ is fully faithful. Fix a pair of nonunital monoids $M$ and $M'$, and let $f_{\bullet }: B_{\bullet } M \rightarrow B_{\bullet } M'$ be a morphism of semisimplicial sets. We wish to show that there is a unique nonunital monoid homomorphism $g: M \rightarrow M'$ such that $f_{\bullet }$ can be recovered by applying the functor $B_{\bullet }$ to $g$. Let us abuse notation by identifying $M$ and $M'$ with the sets $B_{1}M$ and $B_{1}M'$, respectively, so that $f_{\bullet }$ determines a function $f_{1}: M \rightarrow M'$. The uniqueness of $g$ is now clear: if $f_{\bullet } = B_{\bullet } g$, then $g$ must coincide with $f_1$ (as a function). To prove existence, we must establish the following:

$(1)$

The function $f_1: M \rightarrow M'$ is a nonunital monoid homomorphism.

$(2)$

The morphism of semisimplicial sets $f_{\bullet }$ is obtained by applying the functor $B_{\bullet }$ to the homomorphism $f_1$.

We first prove $(1)$. Fix a pair of elements $x,y \in M$ and regard the pair $(x,y)$ as a $2$-simplex $\sigma $ of the semisimplicial set $B_{\bullet } M$. Since $f_{\bullet }$ is a morphism of semisimplicial sets, we have

Assertion $(1)$ now follows by allowing $x$ and $y$ to vary. To prove $(2)$, let $f'_{\bullet }: B_{\bullet } M \rightarrow B_{\bullet } M'$ be the morphism of semisimplicial sets determined by the homomorphism $f_1$, and let $\tau $ be an $n$-simplex of $B_{\bullet } M$; we wish to show that $f_{n}(\tau ) = f'_{n}( \tau )$. Since $\tau $ is determined by its $1$-dimensional faces, we can assume without loss of generality that $n = 1$, in which case the result is clear. This completes the proof that the functor $M \mapsto B_{\bullet } M$ is fully faithful.

Now suppose that $S_{\bullet }$ is a semisimplicial set which satisfies condition $(\ast _ n)$ of Proposition 1.3.2.7 for every integer $n \geq 0$, and set $M = S_1$. For every $n$-tuple of elements $(x_ n, x_{n-1}, \cdots , x_1)$ of $M$, condition $(\ast _ n)$ guarantees that there is a unique $n$-simplex $\sigma _{ x_{n}, \cdots , x_1}$ of $S_{\bullet }$ satisfying $\rho _{i}( \sigma ) = x_ i$, where $\rho _{i}: S_{n} \rightarrow S_1 = M$ is the function induced by the inclusion map $[1] \simeq \{ i-1 < i \} \hookrightarrow [n]$. We can then define a multiplication $m: M \times M \rightarrow M$ by the formula $m(x,y) = d^{2}_1( \sigma _{x,y} )$. This multiplication is associative: for every triple of elements $x,y,z \in M$, we compute

It follows that we can regard $M$ as a nonunital commutative monoid. Moreover, for every integer $n \geq 0$, the construction $(x_ n, \cdots , x_1) \mapsto \sigma _{ x_ n, \cdots , x_1}$ determines a bijection $f_{n}: B_{n} M \rightarrow S_{n}$. We will complete the proof by showing that the collection $\{ f_ n \} _{n \geq 0}$ is an isomorphism of semisimplicial sets: that is, that it commutes with the face operators. Fix an integer $n > 0$ and an $n$-simplex $\tau $ of $B_{\bullet } M$; we wish to show that $d^{n}_ i( f_{n}( \tau ) ) = f_{n-1}( d^{n}_ i(\tau ) )$ for $0 \leq i \leq n$. Let us identify $\tau $ with a tuple of elements $(x_{n}, x_ n, \cdots , x_1)$ of $M$; we wish to verify the identity

For $1 \leq j \leq n-1$, let $\rho _{j}: S_{n-1} \rightarrow S_{1} = M$ be defined as above; we can then rewrite the preceding identity as

This follows immediately from the definition of the simplex $\sigma _{ x_ n, x_{n-1}, \cdots , x_1 }$ in the case $j \neq i$, and from the construction of the multiplication $m$ in the case $j = i$. $\square$