1.1.2 Degeneracy Operators
Let $S_{\bullet }$ be a simplicial set. By virtue of Proposition 1.1.1.9, the underlying semisimplicial set is determined by the sequence of sets $\{ S_{n} \} _{n \geq 0}$ together with the face operators $\{ d^{n}_{i}: S_{n} \rightarrow S_{n-1} \} _{0 \leq i \leq n}$. To recover $S_{\bullet }$ as a simplicial set, we need more information.
Construction 1.1.2.1 (Degeneracy Operators). For every pair of integers $0 \leq i \leq n$ we let $\sigma ^{i}_{n}: [n+1] \twoheadrightarrow [n]$ denote the nondecreasing function given by the formula
\[ \sigma ^{i}_{n}( j) = \begin{cases} j & \text{ if } j \leq i \\ j-1 & \text{ if } j > i. \end{cases} \]
If $C_{\bullet }$ is a simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\sigma ^{i}_{n}$ to obtain a morphism from $C_{n}$ to $C_{n+1}$. We will denote this map by $s^{n}_{i}: C_{n} \rightarrow C_{n+1}$ and refer to it as the $i$th degeneracy operator.
Notation 1.1.2.2. Let $S_{\bullet }$ be a simplicial set. Then the degeneracy operator $s^{0}_{0}: S_{0} \rightarrow S_{1}$ carries each vertex $x$ to an edge of $S_{\bullet }$ which we will denote by $\operatorname{id}_{x}$. Note that the vertex $x$ is both the source and target of the edge $\operatorname{id}_{x}$ (see Exercise 1.1.2.7).
Definition 1.1.2.3. Let $S_{\bullet }$ be a simplicial set. We say that an $n$-simplex $\sigma $ of $S_{\bullet }$ is degenerate if it belongs to the image of the degeneracy operator $s^{n-1}_{i}: S_{n-1} \rightarrow S_{n}$ for some integer $0 \leq i < n$. We say that $\sigma $ is nondegenerate if it is not degenerate. In particular, every $0$-simplex of $S_{\bullet }$ is nondegenerate.
Example 1.1.2.4 (Degenerate Edges). Let $S_{\bullet }$ be a simplicial set and let $e$ be an edge of $S_{\bullet }$. Then $e$ is degenerate if and only if it has the form $\operatorname{id}_{x}$, for some vertex $x \in S_{\bullet }$. If this condition is satisfied, then the vertex $x$ is uniquely determined (since it is both the source and target of the edge $e$).
Exercise 1.1.2.7 (Relations Between Face and Degeneracy Operators). Let $C_{\bullet }$ be a simplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face and degeneracy operators of $\operatorname{\mathcal{C}}$ satisfy the following relations:
- $(\ast ')$
For $0 \leq i \leq n$ and $0 \leq j \leq n+1$, we have an equality
\[ d^{n+1}_{j} \circ s^{n}_ i = \begin{cases} s^{n-1}_{i-1} \circ d^{n}_ j & \text{ if } j < i \\ \operatorname{id}_{ C_ n } & \text{ if } j = i \text{ or } j = i + 1 \\ s^{n-1}_{i} \circ d^{n}_{j-1} & \text{ if } j > i+1 \end{cases} \]
(as morphisms from $C_{n}$ to $C_{n}$).
Example 1.1.2.8 (Degenerate $2$-Simplices). Let $S_{\bullet }$ be a simplicial set and let $\sigma $ be a $2$-simplex of $S_{\bullet }$. We say that $\sigma $ is left-degenerate if it has the form $s^{1}_{0}(e)$, for some edge $e: x \rightarrow y$ of $S_{\bullet }$. In this case, the faces of $\sigma $ are depicted in the diagram
\[ \xymatrix { & x \ar [dr]^{e} & \\ x \ar [ur]^{\operatorname{id}_{x}} \ar [rr]^{e} & & y. } \]
We will say that $\sigma $ is right-degenerate if it has the form $s^{1}_{1}(e)$, for some edge $e: x \rightarrow y$ of $S_{\bullet }$; in this case, the faces of $\sigma $ are depicted in the diagram
\[ \xymatrix { & y \ar [dr]^{\operatorname{id}_ y} & \\ x \ar [ur]^{e} \ar [rr]^{e} & & y. } \]
Note that $\sigma $ is degenerate if and only if it is either left-degenerate or right-degenerate.
Exercise 1.1.2.9. Let $S_{\bullet }$ be a simplicial set and let $\sigma $ be a $2$-simplex of $S_{\bullet }$. Show that $\sigma $ is both left-degenerate and right-degenerate if and only if it is constant: that is, it factors as a composition $\Delta ^2 \twoheadrightarrow \Delta ^0 \hookrightarrow S_{\bullet }$ (for a more general statement, see Proposition 1.1.3.8).
Proposition 1.1.2.10. Let $S_{\bullet }$ be a simplicial set and let $\tau \in S_{n}$ be an $n$-simplex of $S_{\bullet }$ for some $n > 0$, which we will identify with a map of simplicial sets $\tau : \Delta ^{n} \rightarrow S_{\bullet }$. The following conditions are equivalent:
- $(1)$
The simplex $\tau $ belongs to the image of the degeneracy operator $s^{n-1}_{i}: S_{n-1} \rightarrow S_{n}$ for some $0 \leq i < n$ (see Construction 1.1.2.1).
- $(2)$
The map $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {f} \Delta ^{n-1} \rightarrow S_{\bullet }$, where $f$ corresponds to a surjective map of linearly ordered sets $[n] \twoheadrightarrow [n-1]$.
- $(3)$
The map $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {f} \Delta ^{m} \rightarrow S_{\bullet }$, where $m < n$ and $f$ corresponds to a surjective map of linearly ordered sets $[n] \twoheadrightarrow [m]$.
- $(4)$
The map $\tau $ factors as a composition $\Delta ^{n} \rightarrow \Delta ^{m} \rightarrow S_{\bullet }$, where $m < n$.
- $(5)$
The map $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {\tau '} \Delta ^{m} \rightarrow S_{\bullet }$, where $\tau '$ is not injective on vertices.
Proof.
The implications $(1) \Leftrightarrow (2) \Rightarrow (3) \Rightarrow (4) \Rightarrow (5)$ are immediate. We will complete the proof by showing that $(5)$ implies $(1)$. Assume that $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {\tau '} \Delta ^{m} \xrightarrow {\sigma '} S_{\bullet }$, where $\tau '$ is not injective on vertices. Then there exists some integer $0 \leq i < n$ satisfying $\tau '(i) = \tau '(i+1)$. It follows that $\tau '$ factors through the map $\sigma ^{i}_{n-1}: \Delta ^{n} \rightarrow \Delta ^{n-1}$ of Construction 1.1.2.1, so that $\tau $ belongs to the image of the degeneracy operator $s^{n-1}_{i}$.
$\square$
We close this section by showing that a simplicial object $C_{\bullet }$ of a category $\operatorname{\mathcal{C}}$ can be recovered from the sequence of objects $\{ C_{n} \} _{n \geq 0}$, together with the face and degeneracy operators given by Constructions 1.1.1.4 and 1.1.2.1 (Proposition 1.1.2.14). We begin by proving a simpler result, which involves only the degeneracy operators.
Notation 1.1.2.12. Let $\operatorname{{\bf \Delta }}_{\operatorname{surj}}$ denote the category whose objects are the linearly ordered sets $[n] = \{ 0 < 1 < \cdots < n \} $ for $n \geq 0$, and whose morphisms are nondecreasing surjective functions $[m] \twoheadrightarrow [n]$.
Proposition 1.1.2.13. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ C_ n \} _{n \geq 0}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then a system of morphisms $\{ s^{n}_{i}: C_{n} \rightarrow C_{n+1} \} _{0 \leq i \leq n}$ can be obtained from a functor $C_{\bullet }: \operatorname{{\bf \Delta }}_{\operatorname{surj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast '')$ of Remark 1.1.2.11. In this case, the functor $C_{\bullet }$ is uniquely determined.
Proof.
We proceed as in the proof of Proposition 1.1.1.9. Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ and a collection of morphisms $\{ \widetilde{\sigma }_{n}^{i}: [n+1] \rightarrow [n] \} _{0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ obtained by imposing the relation
1.10
\begin{eqnarray} \label{equation:relation-semisimplicial-identity2} \widetilde{\sigma }^{j}_{n} \circ \widetilde{\sigma }^{i}_{n+1} & = & \widetilde{\sigma }^{i}_{n} \circ \widetilde{\sigma }^{j+1}_{n+1} \end{eqnarray}
for every triple of integers $0 \leq i \leq j \leq n$. Using Remark 1.1.2.11, we see that there is a unique functor $F_{\operatorname{surj}}: \overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}} \rightarrow \operatorname{{\bf \Delta }}_{\operatorname{surj}}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ to itself, and each generating morphism $\widetilde{\sigma }_{n}^{i}$ to the epimorphism $\sigma _{n}^{i}: [n+1] \twoheadrightarrow [n]$ of Construction 1.1.2.1. To prove Proposition 1.1.2.13, it will suffice to show that the functor $F_{\operatorname{surj}}$ is an isomorphism of categories.
Fix integers $0 \leq m \leq n$, and set $b = n-m+1$. In the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$, every morphism $\beta : [n] \rightarrow [m]$ admits a unique factorization $\beta = \widetilde{\sigma }_{m}^{i_0} \circ \widetilde{\sigma }_{m+1}^{i_1} \circ \cdots \circ \widetilde{\sigma }_{m+b}^{i_{b}}$, where the superscripts are nonnegative integers satisfying $0 \leq i_ a \leq m+a$ for $0 \leq a \leq b$. Let us say that $\beta $ is in standard form if, in addition, the integers $i_ a$ satisfy the inequalities $i_0 < i_1 < i_2 < \cdots < i_ b$. Note that, by repeatedly applying the relation (1.10), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ to a morphism which is in standard form. More precisely, every morphism $\overline{\beta }: [n] \rightarrow [m]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ which is in standard form.
By construction, the functor $F_{\operatorname{surj}}$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [n] \twoheadrightarrow [m]$ in $\operatorname{{\bf \Delta }}_{\operatorname{surj}}$, there is a unique morphism $\overline{\beta }: [n] \rightarrow [m]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ satisfying $F_{\operatorname{surj}}( \overline{\beta } ) = \alpha $. By virtue of the preceding discussion, it will suffice to show that $\alpha $ can be lifted uniquely to a morphism $\beta : [n] \rightarrow [m]$ in the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ which is in standard form. We now observe that $\beta =\widetilde{\sigma }_{m}^{i_0} \circ \widetilde{\sigma }_{m+1}^{i_1} \circ \cdots \circ \widetilde{\sigma }_{m+b}^{i_{b}}$ is characterized by the requirement that $\{ i_0 < i_1 < \cdots < i_ b \} $ is the collection of integers $0 \leq j < n$ satisfying $\alpha (j) = \alpha (j+1)$.
$\square$
Proposition 1.1.2.14. Let $\operatorname{\mathcal{C}}$ be a category containing a sequence of objects $\{ C_{n} \} _{n \geq 0}$. Then morphisms
\[ \{ d^{n}_{i}: C_{n} \rightarrow C_{n-1} \} _{ 0 \leq i \leq n, n > 0} \quad \quad \{ s^{n}_{i}: C_{n} \rightarrow C_{n+1} \} _{0 \leq i \leq n} \]
are the face and degeneracy operators for a simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast )$ of Remark 1.1.1.7, condition $(\ast ')$ of Exercise 1.1.2.7, and condition $(\ast '')$ of Remark 1.1.2.11. In this case, the simplicial object $C_{\bullet }$ is uniquely determined.
Proof.
We proceed as in the proofs of Propositions 1.1.1.9 and 1.1.2.13. Let $\widetilde{\operatorname{{\bf \Delta }}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ together with morphisms $\{ \widetilde{\delta }_{n}^{i}: [n-1] \rightarrow [n] \} _{n > 0, 0 \leq i \leq n}$ and $\{ \widetilde{\sigma }_{n}^{i}: [n+1] \rightarrow [n] \} _{0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}$ obtained by imposing the relations (1.4) and (1.10), together with the following:
1.11
\begin{eqnarray} \label{equation:relation-semisimplicial-identity3} \widetilde{\sigma }^{j}_{n} \circ \widetilde{\delta }_{n+1}^{i} & = & \begin{cases} \widetilde{\delta }_{n}^{i} \circ \widetilde{\sigma }^{j-1}_{n-1} & \text{ if } i < j \\ \operatorname{id}_{ [n] } & \text{ if } i = j \text{ or } i = j + 1 \\ \widetilde{\delta }^{i-1}_{n} \circ \widetilde{\sigma }^{j}_{n-1} & \text{ if } i > j+1. \end{cases}\end{eqnarray}
for every triple of integers $0 \leq i , j \leq n$. There is a unique functor $F: \overline{\operatorname{{\bf \Delta }}} \rightarrow \operatorname{{\bf \Delta }}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}$ to itself and satisfies $F( \widetilde{\delta }_{n}^{i} ) = \delta _{n}^{i}$ and $F( \widetilde{\sigma }_{n}^{i} ) = \sigma _{n}^{i}$. To prove Proposition 1.1.2.14, it will suffice to show that the functor $F$ is an isomorphism of categories.
Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ be the categories appearing in the proofs of Proposition 1.1.1.9 and Proposition 1.1.2.13, respectively. Let us identify $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ with (non-full) subcategories of $\widetilde{\operatorname{{\bf \Delta }}}$. We will say that a morphism $\beta : [m] \rightarrow [n]$ of $\widetilde{\operatorname{{\bf \Delta }}}$ is weakly standard if it factors as a composition $[m] \xrightarrow { \beta _{\operatorname{surj}} } [k] \xrightarrow { \beta _{\operatorname{inj}} } [n]$, where $\beta _{\operatorname{inj}}$ belongs to $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ belongs to $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$. In this case, the morphisms $\beta _{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ are uniquely determined. We will say that $\beta $ is in standard form if it is weakly standard and, in addition, the morphisms $\beta _{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ are in standard form (as in the proofs of Propositions 1.1.1.9 and 1.1.2.13). Note that, by repeatedly applying the relation (1.11), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}$ into a morphism $\beta $ which is weakly standard. Using the relations (1.4) and (1.10), we can further arrange that $\beta $ is in standard form. It follows that every morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ of $\widetilde{\operatorname{{\bf \Delta }}}$ which is in standard form.
By construction, the functor $F$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, there is a unique morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}$ satisfying $F( \overline{\beta } ) = \alpha $. Let $\widetilde{F}$ denote the composite functor $\widetilde{\operatorname{{\bf \Delta }}} \twoheadrightarrow \overline{\operatorname{{\bf \Delta }}} \xrightarrow {F} \operatorname{{\bf \Delta }}$. By virtue of the preceding discussion, it will suffice to show that there is a unique morphism $\beta : [m] \rightarrow [n]$ in $\widetilde{ \operatorname{{\bf \Delta }}}$ which is in standard form and satisfies $\widetilde{F}( \beta ) = \alpha $. In the simplex category $\operatorname{{\bf \Delta }}$, the morphism $\alpha $ factors uniquely as a composition $[m] \xrightarrow { \alpha _{\operatorname{surj}} } [k] \xrightarrow { \alpha _{\operatorname{inj}} } [n]$, where $\alpha _{\operatorname{inj}}$ is an injection and $\alpha _{\operatorname{surj}}$ is a surjection. If $\beta : [m] \rightarrow [n]$ is a weakly standard morphism of $\widetilde{\operatorname{{\bf \Delta }}}$, then the identity $\widetilde{F}( \beta ) = \alpha $ holds if and only if $\widetilde{F}( \beta _{\operatorname{inj}} ) = \alpha _{\operatorname{inj}}$ and $\widetilde{F}( \beta _{\operatorname{surj}} ) = \alpha _{\operatorname{surj}}$. We are therefore reduced to proving that $\alpha _{\operatorname{inj}}$ and $\alpha _{\operatorname{surj}}$ can be lifted uniquely to morphisms of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ which are in standard form, which was established in the proofs of Proposition 1.1.1.9 and Proposition 1.1.2.13.
$\square$