# Kerodon

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### 1.3.2 Example: Monoids as Simplicial Sets

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

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

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 1.3.2.2. 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

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

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 1.3.2.3. 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

$f(e) = e' \quad \quad f( xy ) = f(x) f(y)$

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.

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

$\{ \text{Categories \operatorname{\mathcal{C}} with \operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X\} } \} \xrightarrow {\sim } \{ \text{Monoids} \} .$

More precisely, there is a pullback diagram of categories

$\xymatrix@C =50pt{ \operatorname{Mon}\ar [r]^-{ M \mapsto BM } \ar [d] & \operatorname{Cat}\ar [d]^{ \operatorname{Ob}} \\ \{ \ast \} \ar [r] & \operatorname{Set}, }$

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

$\operatorname{Ob}( BM ) = \{ X \} \quad \quad \operatorname{Hom}_{BM}(X,X) = M.$

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

$\alpha _{i,i} = e \quad \quad \alpha _{k,i} = \alpha _{k,j} \alpha _{j,i} \text{ for 0 \leq i \leq j \leq k \leq n.}$

For each $n \geq 0$, the construction

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

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

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

(see Remarks 1.3.1.7 and 1.3.1.8).

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

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

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

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 1.3.2.9. 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

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Monoid homomorphisms f: M^{+} \rightarrow M'} \} \ar [d] \\ \{ \textnormal{Nonunital monoid homomorphisms f_0: M \rightarrow M'} \} . }$

Consequently, the inclusion functor $\operatorname{Mon}\hookrightarrow \operatorname{Mon}^{\operatorname{nu}}$ has a left adjoint, given on objects by the construction $M \mapsto M^{+}$.

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

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

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

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

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:

$(1)$

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

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

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 1.3.2.7 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

\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 1.3.2.7. 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
$$\label{equation:monoids-as-semisimplicial-objects} s^{n}_ i( \sigma _{x_{n}, \cdots , x_1} ) = \sigma _{ x_ n, \cdots , x_{i+1}, e, x_ i, \cdots , x_1}.$$

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). $\square$