We now specialize Construction 1.3.1.1 to categories having a single object.
Definition 1.3.2.1. A monoid is a set $M$ equipped with a multiplication map which satisfies the following conditions:
The multiplication $m$ is associative. That is, we have $x(yz) = (xy)z$ for each triple of elements $x,y,z \in M$.
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 $M$).
\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]
Monoids are ubiquitous in mathematics:
Example 1.3.2.2. 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 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 $X$.
\[ \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \times \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \quad \quad (f,g) \mapsto f \circ g, \]
The collection of monoids can be organized into a category:
Definition 1.3.2.3. 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 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.
\[ f(e) = e' \quad \quad f( xy ) = f(x) f(y) \]
Remark 1.3.2.4. The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ of Example 1.3.2.2 induces an equivalence More precisely, there is a pullback diagram of categories where $\ast = \{ X \} $ is the set having a single element $X$. Here the upper horizontal functor assigns to each monoid $M$ the category $BM$ of Construction 1.3.2.5, given concretely by
\[ \{ \text{Categories $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X\} $} \} \xrightarrow {\sim } \{ \text{Monoids} \} . \]
\[ \xymatrix@C =50pt{ \operatorname{Mon}\ar [r]^-{ M \mapsto BM } \ar [d] & \operatorname{Cat}\ar [d]^{ \operatorname{Ob}} \\ \{ \ast \} \ar [r] & \operatorname{Set}, } \]
\[ \operatorname{Ob}( BM ) = \{ X \} \quad \quad \operatorname{Hom}_{BM}(X,X) = M. \]
Construction 1.3.2.5. Let $M$ be a monoid. We let $B_{\bullet }M$ denote the nerve of the category $BM$ described in Remark 1.3.2.4. We will refer to $B_{\bullet }M$ as the classifying simplicial set of the monoid $M$.
Remark 1.3.2.6. Let $M$ be a monoid with unit element $e$ and let $B_{\bullet }M$ denote its classifying simplicial set. By definition, $n$-simplices of the simplicial set $B_{\bullet }M$ are functors from the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ to the category $BM$. Such a functor can be identified with a collection of elements $\{ \alpha _{j,i} \in M \} _{0 \leq i \leq j \leq n}$ (where $\alpha _{j,i}$ denotes the image in $BM$ of the unique element of $\operatorname{Hom}_{[n]}(i,j)$) which are required to satisfy the identities For each $n \geq 0$, the construction induces a bijection $B_{n} M \simeq M^ n$. Under the resulting identification, the face and degeneracy operators of $B_{\bullet }M$ are given concretely by the formulae
\[ \alpha _{i,i} = e \quad \quad \alpha _{k,i} = \alpha _{k,j} \alpha _{j,i} \text{ for $0 \leq i \leq j \leq k \leq n$.} \]
\[ \{ \alpha _{j,i} \} _{0 \leq i \leq j \leq n} \mapsto ( \alpha _{n,n-1}, \alpha _{n-1,n-2}, \cdots , \alpha _{1,0} ) \]
\[ d^{n}_ i( x_ n, x_{n-1}, \ldots , x_1) = \begin{cases} (x_ n, x_{n-1}, \ldots , x_2) & \text{ if } i = 0 \\ (x_ n, \ldots , x_{i+2}, x_{i+1} x_ i, x_{i-1}, \ldots , x_1) & \text{ if } 0 < i < n \\ ( x_{n-1}, x_{n-2}, \ldots , x_1) & \text{ if } i = n \end{cases} \]
\[ s^{n}_ i( x_ n, x_{n-1}, \ldots , x_1 ) = (x_ n, \ldots , x_{i+1}, e, x_{i}, \ldots , x_1) \]
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 $n \geq 0$:
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 1.3.2.7 at the end of this section. As a first step, we establish a simpler result in the setting of semisimplicial sets.
Variant 1.3.2.8. A nonunital monoid is a set $M$ equipped with a map 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.
\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]
Warning 1.3.2.9. The terminology of Variant 1.3.2.8 is not standard. Many authors use the term semigroup for what we call a nonunital monoid.
Remark 1.3.2.10. The category $\operatorname{Mon}$ of monoids (Definition 1.3.2.1) can be regarded as a subcategory of the category $\operatorname{Mon}^{\operatorname{nu}}$ of nonunital monoids (Variant 1.3.2.8). Beware that this subcategory is not full. If $M$ and $M'$ are monoids containing unit elements $e$ and $e'$, respectively, then a nonunital monoid homomorphism $f: M \rightarrow M'$ need not satisfy the identity $f(e) = e'$.
Remark 1.3.2.11. Let $M$ be a nonunital monoid, and let $M^{+} = M \cup \{ e\} $ be the enlargement of $M$ obtained by formally adjoining a new element $e$. Then the multiplication on $M$ extends uniquely to a monoid structure on $M^{+}$ having unit element $e$. Moreover, if $M'$ is any other monoid, then the restriction map $f \mapsto f|_{M}$ induces a bijection Consequently, the inclusion functor $\operatorname{Mon}\hookrightarrow \operatorname{Mon}^{\operatorname{nu}}$ has a left adjoint, given on objects by the construction $M \mapsto M^{+}$.
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Monoid homomorphisms $f: M^{+} \rightarrow M'$} \} \ar [d] \\ \{ \textnormal{Nonunital monoid homomorphisms $f_0: M \rightarrow M'$} \} . } \]
Variant 1.3.2.12. 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 1.3.2.6, the construction induces an identification $B_{n} M \simeq M^ n$. Under this identification, the face operators of $B_{\bullet } M$ are given by the formula
\[ \{ \alpha _{j,i} \} _{0 \leq i < j \leq n} \mapsto ( \alpha _{n,n-1}, \alpha _{n-1,n-2}, \cdots , \alpha _{1,0} ) \]
\[ d^{n}_ i( x_ n, x_{n-1}, \ldots , x_1) = \begin{cases} (x_ n, x_{n-1}, \ldots , x_2) & \text{ if } i = 0 \\ (x_ n, \ldots , x_{i+2}, x_{i+1} x_ i, x_{i-1}, \ldots , x_1) & \text{ if } 0 < i < n \\ ( x_{n-1}, x_{n-2}, \ldots , x_1) & \text{ if } i = n. \end{cases} \]
Remark 1.3.2.13. Construction 1.3.2.5 and Variant 1.3.2.12 are compatible: if $M$ is a monoid and $B_{\bullet } M$ is the classifying simplicial set of Construction 1.3.2.5, then the underlying semisimplicial set of $B_{\bullet } M$ is given by Variant 1.3.2.12.
Proposition 1.3.2.7 has the following nonunital counterpart:
Proposition 1.3.2.14. 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 1.3.2.7, 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 $f_{1}: M \rightarrow M'$. 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:
The function $f_1: M \rightarrow M'$ is a nonunital monoid homomorphism.
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 $j = i$. $\square$
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
Specializing to the case $i=n=1$ and applying the face operator $d^{1}_1$, we obtain an equality
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). $\square$