### 1.2.2 Example: Monoids as Simplicial Sets

We now specialize Construction 1.2.1.1 to categories having a single object.

Definition 1.2.2.1. A *monoid* is a set $M$ equipped with a map Recall that a *monoid* is a set $M$ equipped with a map

\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]

which satisfies the following conditions:

- $(a)$
The multiplication $m$ is associative. That is, we have $x(yz) = (xy)z$ for each triple of elements $x,y,z \in M$.

- $(b)$
There exists an element $e \in M$ such that $ex=x=xe$ for each $x \in M$ (in this case, the element $e$ is uniquely determined; we refer to it as the *unit element* of $M$).

Monoids are ubiquitous in mathematics:

Example 1.2.2.2. Let $\operatorname{\mathcal{C}}$ be a category and let $X$ be an object of $\operatorname{\mathcal{C}}$. An *endomorphism of $X$* is a morphism from $X$ to itself in the category $\operatorname{\mathcal{C}}$. We let $\operatorname{End}_{\operatorname{\mathcal{C}}}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ denote the set of all endomorphisms of $X$. The composition law on $\operatorname{\mathcal{C}}$ determines a map

\[ \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \times \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \quad \quad (f,g) \mapsto f \circ g, \]

which exhibits $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ as a monoid; the unit element of $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ is the identity morphism $\operatorname{id}_{X}: X \rightarrow X$. We refer to $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ as the *endomorphism monoid of $X$*.

The collection of monoids can be organized into a category:

Definition 1.2.2.3. Let $M$ and $M'$ be monoids having unit elements $e$ and $e'$, respectively. A function $f: M \rightarrow M'$ is a *monoid homomorphism* if it satisfies the identities

\[ f(e) = e' \quad \quad f( xy ) = f(x) f(y) \]

for every pair of elements $x,y \in M$. We let $\operatorname{Mon}$ denote the category whose objects are monoids and whose morphisms are monoid homomorphisms.

Construction 1.2.2.5. Let $M$ be a monoid. We let $B_{\bullet }M$ denote the nerve of the category $BM$ described in Remark 1.2.2.4. We will refer to $B_{\bullet }M$ as the *classifying simplicial set* of the monoid $M$.

Proposition 1.2.2.7. 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 $S_{n} \rightarrow \prod _{1 \leq i \leq n} S_1$.

We will give the proof of Proposition 1.2.2.7 at the end of this section. As a first step, we establish a simpler result in the setting of semisimplicial sets.

Variant 1.2.2.8. A *nonunital monoid* is a set $M$ equipped with a map

\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]

which satisfies the associative law $x(yz) = (xy)z$ for $x,y,z \in M$. If $M$ and $M'$ are nonunital monoids, a function $f: M \rightarrow M'$ is a *nonunital monoid homomorphism* if it satisfies the equation $f(xy) = f(x) f(y)$ for every pair of elements $x,y \in M$. We let $\operatorname{Mon}^{\operatorname{nu}}$ denote the category whose objects are nonunital monoids and whose morphisms are nonunital monoid homomorphisms.

Warning 1.2.2.9. The terminology of Variant 1.2.2.8 is not standard. Many authors use the term *semigroup* for what we call a *nonunital monoid*.

Variant 1.2.2.12. Let $M$ be a nonunital monoid. We let $B_{\bullet } M$ denote the semisimplicial set which assigns to each object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}$ the collection of tuples $\{ \alpha _{j,i} \in M \} _{0 \leq i < j \leq n}$ which satisfy the identity $\alpha _{k,i} = \alpha _{k,j} \alpha _{j,i}$ for $0 \leq i < j < k \leq n$. As in Remark 1.2.2.6, the construction

\[ \{ \alpha _{j,i} \} _{0 \leq i < j \leq n} \mapsto ( \alpha _{n,n-1}, \alpha _{n-1,n-2}, \cdots , \alpha _{1,0} ) \]

induces an identification $B_{n} M \simeq M^ n$. Under this identification, the face operators of $B_{\bullet } M$ are given by the formula

\[ d^{n}_ i( x_ n, x_{n-1}, \ldots , x_1) = \begin{cases} (x_ n, x_{n-1}, \ldots , x_2) & \text{ if } i = 0 \\ (x_ n, \ldots , x_{i+2}, x_{i+1} x_ i, x_{i-1}, \ldots , x_1) & \text{ if } 0 < i < n \\ ( x_{n-1}, x_{n-2}, \ldots , x_1) & \text{ if } i = n. \end{cases} \]

Proposition 1.2.2.7 has the following nonunital counterpart:

Proposition 1.2.2.14. The construction $M \mapsto B_{\bullet }M$ determines a fully faithful functor from the category $\operatorname{Mon}^{\operatorname{nu}}$ of nonunital monoids to the category of semisimplicial sets. The essential image of this functor consists of those semisimplicial sets which satisfy condition $(\ast _ n)$ of Proposition 1.2.2.7, for each $n \geq 0$.

**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 $M \rightarrow M'$. The uniqueness of $g$ is now clear: if $f_{\bullet } = B_{\bullet } g$, then $g$ must coincide with $f_1$ (as a map of sets). To prove existence, we must establish the following:

- $(1)$
The function $f: M \rightarrow M'$ 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

\[ f_{1}(xy) = f_1( d^{2}_1( \sigma ) ) = d^{2}_1( f_2(\sigma ) ) = f_1(x) f_1(y). \]

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 1.2.2.7 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

\begin{eqnarray*} m( m(x,y), z) & = & m( d^{2}_1( \sigma _{x,y}), z) \\ & = & d^{2}_1( \sigma _{d^{2}_1( \sigma _{x,y} ), z}) \\ & = & d^{2}_1( d^{3}_2( \sigma _{x,y,z} ) ) \\ & = & d^{2}_1( d^{3}_1( \sigma _{x,y,z} ) ) \\ & = & d^{2}_1( \sigma _{ x, d^{2}_1( \sigma _{y,z} ) } ) \\ & = & m( x, d^{2}_1( \sigma _{y,z}) ) \\ & = & m( x, m(y,z) ). \end{eqnarray*}

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

\[ d^{n}_ i( \sigma _{x_{n}, x_{n-1}, \cdots , x_1}) = \begin{cases} \sigma _{x_{n}, x_{n-1}, \cdots , x_2}) & \text{ if } i = 0 \\ \sigma _{x_{n}, \cdots , x_{i+2}, m(x_{i+1}, x_ i), x_{i-1}, \cdots , x_1} & \text{ if } 0 < i < n \\ \sigma _{x_{n-1}, \cdots , x_1} & \text{ if } i = n. \end{cases} \]

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

\[ \rho _{j}( d^{n}_ i( \sigma _{ x_{n}, x_{n-1}, \cdots , x_1} ) ) = \begin{cases} x_ j & \text{ if } j < i \\ m(x_{j+1}, x_{j}) & \text{ if } j = i \\ x_{j+1} & \text{ if } j > i. \end{cases} \]

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 $j = i$.
$\square$

**Proof of Proposition 1.2.2.7.**
We first show that Construction 1.2.2.5 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 1.2.2.14 (together with Remark 1.2.2.13), 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 1.2.2.14, 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 1.2.2.14. 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

1.28
\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 identities (1.28).
$\square$