Notation 1.1.1.1. Let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ denote the category whose objects are linearly ordered sets of the form $[n] = \{ 0 < 1 < \cdots < n \} $ (where $n$ is a nonnegative integer) and whose morphisms are strictly increasing functions $\alpha : [m] \hookrightarrow [n]$.
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1.1.1 Face Operators
For some applications, it is useful to work with variant of Definition 1.1.0.4.
Definition 1.1.1.2. Let $\operatorname{\mathcal{C}}$ be a category. A semisimplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. We typically use the notation $C_{\bullet }$ to indicate a semisimplicial object of $\operatorname{\mathcal{C}}$, whose value on an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}$ we denote by $C_{n}$. A semisimplicial set is a semisimplicial object of the category of sets.
Remark 1.1.1.3. The category $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ of Notation 1.1.1.1 can be regarded as a (non-full) subcategory of the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.0.2. Consequently, any simplicial object $C_{\bullet }$ of a category $\operatorname{\mathcal{C}}$ has an underlying semisimplicial object, given by the composition We will often abuse notation by identifying a simplicial object of $\operatorname{\mathcal{C}}$ with its underlying semisimplicial object.
\[ \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { C_{\bullet } } \operatorname{\mathcal{C}}. \]
The goal of this section is to make Definition 1.1.1.2 more concrete.
Construction 1.1.1.4 (Face Operators). Let $n$ be a positive integer. For $0 \leq i \leq n$, we let $\delta ^{i}_{n}: [n-1] \rightarrow [n]$ denote the unique strictly increasing function whose image does not contain the element $i$, given concretely by the formula If $C_{\bullet }$ is a (semi)simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\delta ^{i}_{n}$ to obtain a morphism from $C_{n}$ to $C_{n-1}$. We will denote this morphism by $d^{n}_{i}: C_{n} \rightarrow C_{n-1}$ and refer to it as the $i$th face operator.
\[ \delta ^{i}_{n}( j) = \begin{cases} j & \text{ if } j < i \\ j+1 & \text{ if } j \geq i. \end{cases} \]
Example 1.1.1.5. Let $n$ be a positive integer and let $S_{\bullet }$ be a simplicial set. For $0 \leq i \leq n$, the face operator $d^{n}_{i}$ of Construction 1.1.1.4 carries each $n$-simplex $\sigma $ of $S_{\bullet }$ to an $(n-1)$-simplex $d^{n}_{i}(\sigma )$, which we will refer to as the $i$th face of $\sigma $.
Example 1.1.1.6. Let $S_{\bullet }$ be a simplicial set and let $e \in S_{\bullet }$ be an edge of $S_{\bullet }$. Then $s = d^{1}_{1}(e)$ is a vertex of $S_{\bullet }$ which we refer to as the source of $e$, and $t = d^{1}_{0}(e)$ is a vertex of $S_{\bullet }$ which we refer to as the target of $e$. We will sometimes write $e: x \rightarrow y$ to indicate that $e$ is an edge of $S_{\bullet }$ having source vertex $x$ and target vertex $y$.
Remark 1.1.1.7 (Relations Among Face Operators). Let $n \geq 2$ be an integer. For every pair of integers $0 \leq i < j \leq n$, the diagram of linearly ordered sets is commutative: both the clockwise and counterclockwise compositions can be identified with the unique order-preserving bijection $[n-2] \simeq [n] \setminus \{ i < j \} $. It follows that, if $C_{\bullet }$ is a semisimplicial object of a category $\operatorname{\mathcal{C}}$, then the face operators of $C_{\bullet }$ satisfy the following condition:
For $0 \leq i < j \leq n$, we have $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as morphisms from $C_{n}$ to $C_{n-2}$).
\[ \xymatrix@R =50pt@C=50pt{ [n-2] \ar [r]^-{\delta ^{i}_{n-1}} \ar [d]^{ \delta ^{j-1}_{n-1} } & [n-1] \ar [d]^{ \delta ^{j}_{n}} \\[n-1] \ar [r]^-{ \delta ^{i}_{n} } & [n] } \]
Example 1.1.1.8. Let $S_{\bullet }$ be a simplicial set and let $\sigma $ be a $2$-simplex of $S_{\bullet }$. Then $\sigma $ has three faces: the edges $f = d^{2}_{2}(\sigma )$, $g = d^{2}_{0}(\sigma )$, and $h = d^{2}_{1}(\sigma )$. In this case, Remark 1.1.1.7 asserts the following:
The edges $f$ and $h$ have the same source vertex $x \in S_{\bullet }$.
The edges $g$ and $h$ have the same target vertex $z \in S_{\bullet }$.
The target of $f$ and the source of $g$ are the same vertex $y \in S_{\bullet }$.
These relationships can be encoded visually in the diagram
\[ \xymatrix@C =50pt@R=50pt{ & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z. } \]
Remark 1.1.1.7 admits the following converse:
Proposition 1.1.1.9. 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 $\{ d^{n}_{i}: C_{n} \rightarrow C_{n-1} \} _{0 \leq i \leq n, n > 0}$ arise as the face operators of a semisimplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast )$ of Remark 1.1.1.7. Moreover, if this condition is satisfied, then $C_{\bullet }$ is uniquely determined.
Proof. Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ and a collection of morphisms $\{ \widetilde{\delta }_{n}^{i}: [n-1] \rightarrow [n] \} _{n > 0, 0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ obtained by imposing the relation
1.4
\begin{eqnarray} \label{equation:relation-semisimplicial-identity} \widetilde{\delta }^{j}_{n} \circ \widetilde{\delta }_{n-1}^{i} & = & \widetilde{\delta }_{n}^{i} \circ \widetilde{\delta }_{n-1}^{j-1} \end{eqnarray}
for every integer $n \geq 2$ and every pair $0 \leq i < j \leq n$. Using Remark 1.1.1.7, we see that there is a unique functor $F_{\operatorname{inj}}: \overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}} \rightarrow \operatorname{{\bf \Delta }}_{\operatorname{inj}}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ to itself, and each generating morphism $\widetilde{\delta }_{n}^{i}$ to the monomorphism $\delta _{n}^{i}: [n-1] \hookrightarrow [n]$ of Construction 1.1.1.4. To prove Proposition 1.1.1.9, it will suffice to show that the functor $F_{\operatorname{inj}}$ is an isomorphism of categories.
Fix integers $0 \leq m \leq n$, and set $b = n-m-1$. In the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$, every morphism $\beta : [m] \rightarrow [n]$ admits a unique factorization $\beta = \widetilde{\delta }_{n}^{i_0} \circ \widetilde{\delta }_{n-1}^{i_1} \circ \cdots \circ \widetilde{\delta }_{n-b}^{i_{b}}$, where the superscripts are nonnegative integers satisfying $0 \leq i_ a \leq n - a$ 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 (1.4), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ to a morphism which is in standard form. More precisely, every morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ which is in standard form.
By construction, the functor $F_{\operatorname{inj}}$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [m] \hookrightarrow [n]$, there is a unique morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ satisfying $F_{\operatorname{inj}}( \overline{\beta } ) = \alpha $. By virtue of the preceding discussion, it will suffice to show that $\alpha $ can be lifted uniquely to a morphism $\beta : [m] \rightarrow [n]$ in the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ which is in standard form. We now observe that $\beta = \widetilde{\delta }_{n}^{i_0} \circ \widetilde{\delta }_{n-1}^{i_1} \circ \cdots \circ \widetilde{\delta }_{n-b}^{i_{b}}$ is characterized by the requirement that $\{ i_ b < i_{b-1} < \cdots < i_0 \} \subseteq [n]$ is the complement of the image of $\alpha $. $\square$