Definition 3.3.1.1. Let $X_{\bullet }$ be a simplicial set. We will say that $X_{\bullet }$ is braced if, for every nondegenerate simplex $\sigma \in X_{n}$ of dimension $n > 0$, the faces $\{ d^{n}_ i(\sigma ) \} _{0 \leq i \leq n}$ are also nondegenerate.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
3.3.1 Digression: Braced Simplicial Sets
Let $\operatorname{{\bf \Delta }}$ denote the simplex category (Definition 1.1.0.2), and let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ denote the subcategory of $\operatorname{{\bf \Delta }}$ spanned by the injective maps (Definition 1.1.1.2). Composition with the inclusion functor $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}}$ determines a forgetful functor from the category $\operatorname{Set_{\Delta }}= \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set})$ of simplicial sets to the category $\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj}}, \operatorname{Set})$ of semisimplicial sets (Remark 1.1.1.3). Our goal in this section is to show that this functor admits a faithful left adjoint, which we will denote by $S_{\bullet } \mapsto S_{\bullet }^{+}$. We begin by describing the essential image of this left adjoint.
Exercise 3.3.1.2. Let $\operatorname{\mathcal{C}}$ be a category. Show that the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is braced if and only if $\operatorname{\mathcal{C}}$ satisfies the following condition:
For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow X$ in $\operatorname{\mathcal{C}}$ satisfying $g \circ f = \operatorname{id}_{X}$, we have $X = Y$ and $f = g = \operatorname{id}_{X}$.
In particular, for any partially ordered set $Q$, the nerve $\operatorname{N}_{\bullet }(Q)$ is braced.
Example 3.3.1.3. Every simplicial set of dimension $\leq 1$ is braced.
Notation 3.3.1.4. Let $X_{\bullet }$ be a simplicial set. For each nonnegative integer $n$, we let $X_{n}^{\mathrm{nd}} \subseteq X_{n}$ denote the collection of nondegenerate $n$-simplices of $X_{\bullet }$. If $X_{\bullet }$ is braced (Definition 3.3.1.1), then the face operators $\{ d^{n}_ i: X_{n} \rightarrow X_{n-1} \} _{0 \leq i \leq n}$ carry $X_{n}^{\mathrm{nd}}$ into $X_{n-1}^{\mathrm{nd}}$. In this case, the construction $[n] \mapsto X_{n}^{\mathrm{nd}}$ determines a semisimplicial set, which we will denote by $X_{\bullet }^{\mathrm{nd}}$.
The terminology of Definition 3.3.1.1 is motivated by the heuristic that a braced simplicial set $X_{\bullet }$ is “supported” by the semisimplicial subset $X_{\bullet }^{\mathrm{nd}} \subseteq X_{\bullet }$. This heuristic is supported by the following:
Proposition 3.3.1.5. Let $X_{\bullet }$ and $Y_{\bullet }$ be simplicial sets, and suppose that $X_{\bullet }$ is braced. Then the restriction map is a bijection.
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Morphisms of simplicial sets $f: X_{\bullet } \rightarrow Y_{\bullet }$} \} \ar [d] \\ \{ \textnormal{Morphisms of semisimplicial sets $f_0: X_{\bullet }^{\mathrm{nd}} \rightarrow Y_{\bullet }$} \} } \]
Proof. Fix a morphism of semisimplicial sets $f_0: X_{\bullet }^{\mathrm{nd} } \rightarrow Y_{\bullet }$; we wish to show that $f_0$ extends uniquely to a morphism of simplicial sets from $X_{\bullet }$ to $Y_{\bullet }$. Let $\sigma $ be an $n$-simplex of $X_{\bullet }$. By virtue of Proposition 1.1.3.8, we can write $\sigma $ uniquely as $\alpha ^{\ast }(\tau )$, where $\alpha : [n] \twoheadrightarrow [m]$ is a nondecreasing surjection and $\tau $ is a nondegenerate $m$-simplex of $X_{\bullet }$. Define $f(\sigma ) = \alpha ^{\ast } f_0(\tau ) \in Y_{n}$. It is clear that any extension of $f_0$ to a morphism of simplicial sets $X_{\bullet } \rightarrow Y_{\bullet }$ must be given by the construction $\sigma \mapsto f(\sigma )$. It will therefore suffice to show that the construction $\sigma \mapsto f(\sigma )$ is a morphism of simplicial sets.
Let $\sigma $, $\tau $, and $\alpha $ be as above, and fix a nondecreasing map $\beta : [n'] \rightarrow [n]$. We wish to prove that $f( \beta ^{\ast } \sigma ) = \beta ^{\ast } f(\sigma )$ in the set $Y_{n'}$. Note that $(\alpha \circ \beta ): [n'] \rightarrow [m]$ factors uniquely as a composition $[n'] \xrightarrow { \alpha ' } [m'] \xrightarrow {\beta '} [m]$, where $\alpha '$ is surjective and $\beta '$ is injective. Since $X_{\bullet }$ is braced, $\beta '^{\ast }(\tau )$ is a nondegenerate $m'$-simplex of $X_{\bullet }$. We now compute
\begin{eqnarray*} f( \beta ^{\ast } \sigma ) & = & f( \beta ^{\ast } \alpha ^{\ast } \tau ) \\ & = & f( \alpha '^{\ast } \beta '^{\ast } \tau ) \\ & = & \alpha '^{\ast } f_0( \beta '^{\ast } \tau ) \\ & = & \alpha '^{\ast } \beta '^{\ast } f_0( \tau ) \\ & = & \beta ^{\ast } \alpha ^{\ast } f_0( \tau ) \\ & = & \beta ^{\ast } f(\sigma ). \end{eqnarray*}
where the second and fifth equality follow from the identity $\alpha \circ \beta = \beta ' \circ \alpha '$, the third and sixth equality follow from the definition of $f$, and the fourth equality from the fact that $f_0$ is a morphism of semisimplicial sets. $\square$
We now show that every semisimplicial set $S_{\bullet }$ can be obtained from the procedure of Notation 3.3.1.4.
Construction 3.3.1.6. Let $S_{\bullet }$ be a semisimplicial set. For each $n \geq 0$, we let $S^{+}_{n}$ denote the collection of pairs $( \alpha , \tau )$ where $\alpha : [n] \twoheadrightarrow [m]$ is a nondecreasing surjection of linearly ordered sets and $\tau $ is an element of $S_{m}$. Let $\beta : [n'] \rightarrow [n]$ be a morphism in the category $\operatorname{{\bf \Delta }}$. For every element $(\alpha , \tau ) \in S^{+}_{n}$, the composite map $\alpha \circ \beta : [n'] \rightarrow [m]$ factors uniquely as a composition $[n'] \xrightarrow {\alpha '} [m'] \xrightarrow {\beta '} [m]$, where $\alpha '$ is surjective and $\beta '$ is injective. We define a map $\beta ^{\ast }: S^{+}_{n} \rightarrow S^{+}_{n'}$ by the formula $\beta ^{\ast }( \alpha , \tau ) = ( \alpha ', \beta '^{\ast }(\tau ) ) \in S_{n'}^{+}$.
Proposition 3.3.1.7. Let $S_{\bullet }$ be a semisimplicial set. Then:
The assignments
of Construction 3.3.1.6 define a simplicial set $S^{+}_{\bullet }$.
The construction $(\tau \in S_{n}) \mapsto ( (\operatorname{id}_{[m]}, \tau ) \in S^{+}_{n} )$ determines a monomorphism of semisimplicial sets $\iota : S_{\bullet } \hookrightarrow S^{+}_{\bullet }$.
The simplicial set $S^{+}_{\bullet }$ is braced, and $\iota $ induces an isomorphism from $S_{\bullet }$ to the semisimplicial subset $(S^{+}_{\bullet })^{\mathrm{nd}} \subseteq S^{+}_{\bullet }$.
\[ ( [n] \in \operatorname{{\bf \Delta }}) \mapsto S^{+}_{n} \quad \quad ( \beta : [n'] \rightarrow [n] ) \mapsto ( \beta ^{\ast }: S^{+}_{n} \rightarrow S^{+}_{n'} ) \]
Proof. It follows immediately that for each $n \geq 0$, the function $\operatorname{id}_{[n]}^{\ast }: S_{n}^{+} \rightarrow S_{n}^{+}$ is the identity map. To prove $(1)$, it will suffice to show that for every pair of composable morphisms $[n''] \xrightarrow {\gamma } [n'] \xrightarrow {\beta } [n]$ in $\operatorname{{\bf \Delta }}$, we have an equality $\gamma ^{\ast } \circ \beta ^{\ast } = ( \beta \circ \gamma )^{\ast }$ of functions from $S_{n}^{+}$ to $S_{n''}^{+}$. Fix an element $(\alpha , \tau ) \in S_{n}^{+}$, where $\alpha : [n] \rightarrow [m]$ is a surjective nondecreasing function and $\tau $ is an element of $S_{m}$. There is a unique commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ [n''] \ar [r]^-{\gamma } \ar [d]^{\alpha ''} & [n'] \ar [d]^{\alpha '} \ar [r]^-{\beta } & [n] \ar [d]^{\alpha } \\ {[m'']} \ar [r]^-{ \gamma ' } & [m'] \ar [r]^-{ \beta ' } & [m] } \]
in the category $\operatorname{{\bf \Delta }}$, where the vertical maps are surjective and the lower horizontal maps are injective. We then compute
\begin{eqnarray*} (\gamma ^{\ast } \circ \beta ^{\ast } )( \alpha , \tau ) & = & \gamma ^{\ast } ( \alpha ', \beta '^{\ast } \tau ) \\ & = & ( \alpha '', \gamma '^{\ast } \beta '^{\ast } \tau ) \\ & = & ( \alpha '', ( \beta ' \circ \gamma ')^{\ast } \tau ) \\ & = & (\beta \circ \gamma )^{\ast } ( \alpha , \tau ), \end{eqnarray*}
which completes the proof of $(1)$.
Assertion $(2)$ is immediate from the definition. Note that if $\beta : [n'] \rightarrow [n]$ is a nondecreasing surjection, then the map $\beta ^{\ast }: S_{n}^{+} \rightarrow S_{n'}^{+}$ is given by the formula $\beta ^{\ast }( \alpha , \tau ) = ( \alpha \circ \beta , \tau )$. It follows that an $n$-simplex $\sigma = (\alpha , \tau )$ of $S_{\bullet }^{+}$ is nondegenerate if and only if $\alpha : [n] \twoheadrightarrow [m]$ is a bijection: that is, if and only if $\sigma $ belongs to the image of $\iota $. Since the image of $\iota $ is closed under face operators (by virtue of $(2)$), we conclude that $S_{\bullet }^{+}$ is braced and that $\iota $ induces an isomorphism of semisimplicial sets $S_{\bullet } \simeq ( S^{+}_{\bullet } )^{\mathrm{nd} }$. $\square$
Corollary 3.3.1.8. Let $\operatorname{Set}_{\Delta }^{\mathrm{br}} \subseteq \operatorname{Set_{\Delta }}$ denote the (non-full) subcategory whose objects are braced simplicial sets and whose morphisms are maps $f: X_{\bullet } \rightarrow Y_{\bullet }$ which carry nondegenerate simplices of $X_{\bullet }$ to nondegenerate simplices of $Y_{\bullet }$. Then the construction $X_{\bullet } \mapsto X_{\bullet }^{\mathrm{nd}}$ induces an equivalence of categories $\operatorname{Set}_{\Delta }^{\mathrm{br}} \rightarrow \{ \textnormal{Semisimplicial sets} \} $, with homotopy inverse given by the construction $S_{\bullet } \rightarrow S_{\bullet }^{+}$.
Proof. Let $X_{\bullet }$ and $Y_{\bullet }$ be braced simplicial sets. It follows from Proposition 3.3.1.5 that the restriction functor $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X_{\bullet }, Y_{\bullet } ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}, \operatorname{Set})}( X_{\bullet }^{\mathrm{nd}}, Y_{\bullet } )$ is a bijection. Moreover, the image of $\operatorname{Hom}_{\operatorname{Set}_{\Delta }^{\mathrm{br}}}( X_{\bullet }, Y_{\bullet } )$ under this bijection is the collection of morphisms of semisimplicial sets from $X_{\bullet }^{\mathrm{nd}}$ to $Y_{\bullet }^{\mathrm{nd}} \subseteq Y_{\bullet }$. This proves full-faithfulness, and the essential surjectivity follows from Proposition 3.3.1.7. $\square$
Corollary 3.3.1.9. Let $S_{\bullet }$ be a semisimplicial set. Then, for every simplicial set $Y_{\bullet }$, composition with the map $\iota : S_{\bullet } \hookrightarrow S^{+}_{\bullet }$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Morphisms of simplicial sets $f: S^{+}_{\bullet } \rightarrow Y_{\bullet }$} \} \ar [d] \\ \{ \textnormal{Morphisms of semisimplicial sets $f_0: S_{\bullet } \rightarrow Y_{\bullet }$} \} . } \]
Proof. Combine Proposition 3.3.1.5 with Proposition 3.3.1.7. $\square$
Corollary 3.3.1.10. The forgetful functor has a left adjoint, given on objects by the construction $S_{\bullet } \mapsto S_{\bullet }^{+}$.
\[ \{ \textnormal{Simplicial sets} \} \rightarrow \{ \textnormal{Semisimplicial sets} \} \]
Corollary 3.3.1.11. Let $X_{\bullet }$ be a braced simplicial set. Then the inclusion of semisimplicial sets $g_0: X_{\bullet }^{\mathrm{nd}} \hookrightarrow X_{\bullet }$ extends uniquely to an isomorphism $g: ( X_{\bullet }^{\mathrm{nd} })^{+} \simeq X_{\bullet }$.
Proof. It follows from Corollary 3.3.1.9 that $g_0$ extends uniquely to a map of simplicial sets $g: ( X_{\bullet }^{\mathrm{nd}} )^{+} \rightarrow X_{\bullet }$. To show that $g$ is an isomorphism, it will suffice to show that for every simplicial set $Y_{\bullet }$, composition with $g$ induces a bijection
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X_{\bullet }, Y_{\bullet } ) & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( (X_{\bullet }^{\mathrm{nd}})^{+}, Y_{\bullet } ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}, \operatorname{Set})}( X_{\bullet }^{\mathrm{nd}}, Y_{\bullet } ), \end{eqnarray*}
which is precisely the content of Proposition 3.3.1.5. $\square$
Example 3.3.1.12. Let $M$ be a nonunital monoid and let $M^{+} = M \cup \{ e\} $ denote the monoid obtained from $M$ by adjoining a unit element (Remark 1.3.2.11). Let $B_{\bullet }( M^{+} )$ denote the classifying simplicial set of Construction 1.3.2.5, and let $B_{\bullet }(M)$ be the semisimplicial set introduced in Variant 1.3.2.12. The inclusion map $M \hookrightarrow M^{+}$ induces a monomorphism of semisimplicial sets $\iota : B_{\bullet }M \hookrightarrow B_{\bullet }(M^{+} )$, whose image consists of the nondegenerate simplices of $B_{\bullet }(M^{+})$. It follows that the simplicial set $B_{\bullet }(M^{+} )$ is braced and that $\iota $ extends to an isomorphism of simplicial sets $(B_{\bullet } M)^{+} \xrightarrow {\sim } B_{\bullet }(M^{+})$ (Corollary 3.3.1.11).