Definition 10.2.2.1 (Semisimplicial Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A semisimplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$. A cosemisimplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \rightarrow \operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
10.2.2 Semisimplicial Objects
Let $\operatorname{{\bf \Delta }}$ denote the simplex category (Definition 1.1.0.2), and let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ denote the subcategory of $\operatorname{{\bf \Delta }}$ whose morphisms are strictly increasing functions $[m] \hookrightarrow [n]$ (Definition 1.1.1.2). It will often be useful to consider the following variant of Definition 10.2.0.1:
Notation 10.2.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often use the notation $X_{\bullet }$ to indicate a semisimplicial object of $\operatorname{\mathcal{C}}$. In this case, we write $X_{n}$ for the value of the functor $X_{\bullet }$ on the object $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj}}$. Similarly, we often use an expression like $X^{\bullet }$ to indicate a cosemisimplicial object of $\operatorname{\mathcal{C}}$, and $X^{n}$ for its value on the object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}$.
Example 10.2.2.3. Let $\operatorname{\mathcal{C}}$ be a category. Then (co)semisimplicial objects of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.2.2.1) can be identified with (co)semisimplicial objects of $\operatorname{\mathcal{C}}$ (in the sense of Definition 1.1.1.2).
Example 10.2.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. The composite functor is a semisimplicial object of $\operatorname{\mathcal{C}}$, which we will refer to as the underlying semisimplicial object of $X_{\bullet }$. We will often abuse notation by identifying $X_{\bullet }$ with its underlying semisimplicial object. Similarly, every cosimplicial object $X^{\bullet }$ of $\operatorname{\mathcal{C}}$ has an underlying cosemisimplicial object, given by the composition
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) \xrightarrow { X^{\bullet } } \operatorname{\mathcal{C}}. \]
Remark 10.2.2.5 (Face Operators). Let $X_{\bullet }$ be a semisimplicial object of an $\infty $-category. For every pair of integers $0 \leq i \leq n$ with $n > 0$, we let $d^{n}_{i}: X_{n} \rightarrow X_{n-1}$ denote the morphism induced by the inclusion of linearly ordered sets $\delta ^{i}_{n}: [n-1] \hookrightarrow [n]$ introduced in Construction 1.1.1.4. We will refer to $d^{n}_{i}$ as the $i$th face operator of the semisimplicial object $X_{\bullet }$.
The utility of Definition 10.2.2.1 stems in part from the fact that, for many purposes, passage from simplicial to semisimplicial objects does not lose very much information.
Proposition 10.2.2.6. The inclusion $\operatorname{{\bf \Delta }}_{\operatorname{inj}} \subset \operatorname{{\bf \Delta }}$ determines a left cofinal functor of $\infty $-categories $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})$.
Proof. By virtue of Theorem 7.2.3.1, it will suffice to show that for every integer $n \geq 0$, the category $\operatorname{\mathcal{C}}= \operatorname{{\bf \Delta }}_{\operatorname{inj}} \times _{\operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{ / [n] }$ has weakly contractible nerve. Let $C_0 \in \operatorname{\mathcal{C}}$ denote the object corresponding to the inclusion map $[0] \simeq \{ n\} \hookrightarrow [n]$. For every object $C \in \operatorname{\mathcal{C}}$, given by a nondecreasing function $\alpha : [m] \rightarrow [n]$, we let $F(C) \in \operatorname{\mathcal{C}}$ denote the object given by the nondecreasing function $\alpha ^{+}: [m+1] \rightarrow [n]$ given by the formula
\[ \alpha ^{+}(i) = \begin{cases} \alpha (i) & \text{ if } 0 \leq i \leq m \\ n & \text{ if } i = m+1. \end{cases} \]
Note that we have canonical maps $C \xrightarrow {\beta _{-}} F(C) \xleftarrow {\beta _+} C_0$, given by the inclusions
\[ \{ 0 < 1 < \cdots < m \} \hookrightarrow \{ 0 < 1 < \cdots < m+1 \} \hookleftarrow \{ m+1 \} . \]
These morphisms depend functorially on $C$, and therefore furnish natural transformations of functors ${\operatorname{id}}_{\operatorname{\mathcal{C}}} \rightarrow F \leftarrow \underline{C}_0$, where $\underline{C}_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denotes the constant functor taking the value $C_0$. It follows that the identity morphism of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is homotopic to the constant morphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \twoheadrightarrow \{ C_0 \} \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, so that the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is contractible (and, in particular, it is weakly contractible). $\square$
Corollary 10.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. Then an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.2.1.3) if and only if it is a colimit of the underlying diagram $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow {X_{\bullet }} \operatorname{\mathcal{C}}$.
Proof. Combine Proposition 10.2.2.6 with Corollary 7.2.2.11. $\square$
Example 10.2.2.8. Let $S$ be a simplicial set and let $| S |$ denote its geometric realization as a topological space (Definition 1.2.3.1). Combining Proposition 10.2.1.1 with Corollary 10.2.2.7, we deduce that the homotopy type of the topological space $| S |$ depends only on the underlying semisimplicial set of $S$. Compare with Corollary 3.4.5.5.
Motivated by Corollary 10.2.2.7, we introduce the following terminology:
Definition 10.2.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a semisimplicial object of $\operatorname{\mathcal{C}}$. We say that an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ if it is a colimit of the diagram $X_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj}} ) \rightarrow \operatorname{\mathcal{C}}$. If $X^{\bullet }$ is a cosemisimplicial object of $\operatorname{\mathcal{C}}$, we say that an object $X \in \operatorname{\mathcal{C}}$ is a totalization of $X^{\bullet }$ if it is a limit of the diagram $X^{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \rightarrow \operatorname{\mathcal{C}}$.
Remark 10.2.2.10. Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. Corollary 10.2.2.7 asserts that an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.2.1.3) if and only if it is a geometric realization of the underlying semisimplicial object of $X_{\bullet }$ (in the sense of Definition 10.2.2.9). In particular, $X_{\bullet }$ admits a geometric realization if and only if its underlying semisimplicial object admits a geometric realization.
In the setting of classical category theory, the notion of geometric realization can be made more concrete.
Proposition 10.2.2.11. Let $\operatorname{\mathcal{C}}$ be a category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $\underline{Y}$ denote the constant semisimplicial object of $\operatorname{\mathcal{C}}$ taking the value $Y$. For every semisimplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, the evaluation map is a monomorphism, whose image is the set of morphisms $\epsilon : X_0 \rightarrow Y$ which satisfy the following condition:
The face operators $d^{1}_0, d^{1}_1: X_1 \rightarrow X_0$ of the simplicial object $X_{\bullet }$ satisfy $\epsilon \circ d^{1}_0 = \epsilon \circ d^{1}_1$.
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj}}, \operatorname{\mathcal{C}}) }( X_{\bullet }, \underline{Y} ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, Y ) \]
Proof. For every integer $n \geq 0$, let $\iota _{n}$ denote the inclusion map $[0] = \{ 0 \} \hookrightarrow \{ 0 < 1 < \cdots < n \} = [n]$ and write $\iota _{n}^{\ast }: X_{n} \rightarrow X_0$ for the associated morphism of $\operatorname{\mathcal{C}}$. If $f_{\bullet }: X_{\bullet } \rightarrow \underline{Y}$ is a morphism of semisimplicial objects, then we must have $f_{n} = f_0 \circ \iota ^{\ast }_{n}$ for each $n \geq 0$; in particular, $f_{\bullet }$ is uniquely determined by the morphism $\epsilon = f_0$. To complete the proof, it will suffice to show that if a morphism $\epsilon : X_0 \rightarrow Y$ satisfies condition $(\ast )$, then the collection $\{ (\epsilon \circ \iota ^{\ast }_ n): X_ n \rightarrow Y \} _{n \geq 0}$ determines a morphism of semisimplicial objects from $X_{\bullet }$ to $\underline{Y}$ (the converse follows immediately from the definitions). Fix a strictly increasing function $\alpha : [m] \hookrightarrow [n]$; we wish to show that the diagram
10.3
\begin{equation} \begin{gathered}\label{equation:maps-into-constant-simplicial-easy} \xymatrix@R =50pt@C=50pt{ X_{n} \ar [d]^{ \alpha ^{\ast } } \ar [r]^-{ \iota ^{\ast }_{n} } & X_{0} \ar [d]^{ \epsilon } \\ X_{m} \ar [r]^-{ \epsilon \circ \iota ^{\ast }_{m} } & Y } \end{gathered} \end{equation}
commutes. If $\alpha (0) = 0$, then $\iota _{n} = \alpha \circ \iota _{m}$. It follows that $\iota _{n}^{\ast } = \iota _{m}^{\ast } \circ \alpha ^{\ast }$, and the desired result follows by composing with $\epsilon $ on both sides. We may therefore assume without loss of generality that $\alpha (0) > 0$. Let $\beta : [1] \hookrightarrow [n]$ be the strictly increasing function given by $\beta (0) = 0$ and $\beta (1) = \alpha (0)$. Then (10.3) can be identified with the outer rectangle of the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{n} \ar [d]^{ \alpha ^{\ast } } \ar [r]^-{ \beta ^{\ast } } & X_{1} \ar [r]^-{ d^{1}_1 } \ar [d]^{d^{1}_0} & X_0 \ar [d]^{ \epsilon } \\ X_{m} \ar [r]^-{ \iota ^{\ast }_ m } & X_0 \ar [r]^-{\epsilon } & Y, } \]
where the left square commutes by the naturality of the construction $[k] \mapsto X_{k}$, and the right square commutes by virtue of assumption $(\ast )$. $\square$
Corollary 10.2.2.12. Let $X_{\bullet }$ be a semisimplicial object of a category $\operatorname{\mathcal{C}}$. Then an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ (in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$) if and only if it is a coequalizer of the face operators $d^{1}_0, d^{1}_1: X_1 \rightrightarrows X_0$.
For some applications, we will need more precise language for discussing geometric realizations of semisimplicial objects.
Notation 10.2.2.13. Let $\operatorname{{\bf \Delta }}_{+}$ be the augmented simplex category (Definition 10.2.1.10). We let $\operatorname{{\bf \Delta }}_{\operatorname{inj},+}$ denote the (non-full) subcategory of $\operatorname{{\bf \Delta }}_{+}$ whose morphisms are strictly increasing functions $[m] \hookrightarrow [n]$. Note that $\operatorname{{\bf \Delta }}_{\operatorname{inj},+}$ can be obtained from the category $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ by adjoining an initial object $[-1]$ satisfying $\operatorname{Hom}_{ \operatorname{{\bf \Delta }}_{+, \operatorname{inj}} }( [n], [-1] ) = \emptyset $ for $n \geq 0$. Consequently, $\operatorname{{\bf \Delta }}_{\operatorname{inj},+}$ can be identified with the left cone $\operatorname{{\bf \Delta }}^{\triangleleft }_{\operatorname{inj}}$ (see Example 4.3.2.5).
Proposition 10.2.2.14. The diagram of $\infty $-categories is a categorical pushout square.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \ar [r] \ar [d] & \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) \ar [d] \\ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}, +} ) \ar [r] & \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+} ) } \]
Proof. By virtue of Proposition 7.2.2.1, this is a reformulation of Proposition 10.2.2.6. $\square$
Definition 10.2.2.15 (Augmented Semisimplicial Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An augmented semisimplicial object of $\operatorname{\mathcal{C}}$ is a functor from the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj},+} )$ to $\operatorname{\mathcal{C}}$. An augmented cosemisimplicial object is a functor from the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+} )$ to $\operatorname{\mathcal{C}}$.
Notation 10.2.2.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often use the notation $X_{\bullet }$ to indicate an augmented semisimplicial object of $\operatorname{\mathcal{C}}$. In this case, we write $X_{n}$ for the value of the functor $X_{\bullet }$ on the object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj},+}^{\operatorname{op}}$. Similarly, we often use the expression $X^{\bullet }$ to indicate an augmented cosimplicial object of $\operatorname{\mathcal{C}}$, and $X^{n}$ for its value on the object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj},+}$.
Remark 10.2.2.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Every augmented semisimplicial object of $\operatorname{\mathcal{C}}$ determines a semisimplicial object of $\operatorname{\mathcal{C}}$, by restriction along the inclusion of full subcategories $\operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj},+}$. For this reason, we will sometimes use the notation $\overline{X}_{\bullet }$ to indicate an augmented semisimplicial object of $\operatorname{\mathcal{C}}$, to distinguish it from the underlying simplicial object $X_{\bullet } = \overline{X}_{\bullet }|_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} )}$.
Remark 10.2.2.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. It follows from Proposition 10.2.2.14 that the diagram of $\infty $-categories is a categorical pullback square. In particular, if $X_{\bullet }$ is a simplicial object of $\operatorname{\mathcal{C}}$, then the datum of an augmentation of $X_{\bullet }$ is equivalent to the datum of an augmentation on the underlying semisimplicial object of $X_{\bullet }$.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj},+}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}), \operatorname{\mathcal{C}})} \]
Remark 10.2.2.19 (Face Operators). For every pair of integers $0 \leq i \leq n$, there is a unique increasing function $\delta _{n}^{i}: [n-1] \hookrightarrow [n]$ whose image is the set $[n] \setminus \{ i\} $, given concretely by the formula If $X_{\bullet }$ is an augmented semisimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$, then evaluation on the morphism $\delta ^{i}_{n}$ determines a map $d^{n}_{i}: X_{n} \rightarrow X_{n-1}$, which we will refer to as the $i$th face operator for the augmented semisimplicial object $X_{\bullet }$. If $n > 0$, this recover the face operators for the underlying semisimplicial object of $X_{\bullet }$ (Remark 10.2.2.5). In the case $n = 0$, we obtain a new operator $d^{0}_{0}: X_{0} \rightarrow X_{-1}$.
\[ \delta ^{i}_{n}( j) = \begin{cases} j & \text{ if } j < i \\ j+1 & \text{ if } j \geq i. \end{cases} \]
Remark 10.2.2.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. By virtue of Remark 10.2.1.11, the following data are equivalent:
Augmented semisimplicial objects $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ carrying the object $[-1]$ to $X$.
Semisimplicial objects of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$.
We will often invoke this equivalence implicitly, using the notation $X_{\bullet }$ to indicate both an augmented simplicial object of $\operatorname{\mathcal{C}}$ (satisfying $X_{-1} = X$) and the associated simplicial object of $\operatorname{\mathcal{C}}_{/X}$.
Remark 10.2.2.21 (Augmented Moore Complexes). Let $A_{\bullet }$ be an augmented semisimplicial object of the category of abelian groups. For each $n \geq 0$, let $\partial : A_{n} \rightarrow A_{n-1}$ denote the group homomorphism given by the alternating sum The diagram is a chain complex of abelian groups which we will denote by $\mathrm{C}^{\operatorname{aug}}_{\ast }(A)$ and refer to as the augmented Moore complex of $A_{\bullet }$. Note that, when restricted to nonnegative degrees, this recovers the Moore complex of the underlying semisimplicial abelian group (see Construction 2.5.5.1).
\[ \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(\sigma ). \]
\[ \cdots \rightarrow A_{2} \xrightarrow {\partial } A_{1} \xrightarrow {\partial } A_{0} \xrightarrow {\partial } A_{-1} \]
Variant 10.2.2.22. Let $A_{\bullet }$ be an augmented simplicial object of the category of abelian groups. Let us abuse notation by identifying $A_{\bullet }$ with the underlying simplicial abelian group, and let be the subcomplex generated by the images of the degeneracy operators (see Proposition 2.5.5.6). We let $\mathrm{N}_{\ast }^{\operatorname{aug}}(A)$ denote the quotient complex $\mathrm{C}^{\operatorname{aug}}_{\ast }(A) / \mathrm{D}_{\ast }(A)$, which we will refer to as the normalized augmented Moore complex of $A_{\bullet }$. Note that, when restricted to nonnegative degrees, this recovers the normalized Moore complex of the underlying simplicial abelian group (Construction 2.5.5.7).
\[ \mathrm{D}_{\ast }(A) \subseteq \mathrm{C}_{\ast }(A) \subseteq \mathrm{C}_{\ast }^{\operatorname{aug}}( A ) \]
Definition 10.2.2.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\overline{X}_{\bullet }$ be an augmented semisimplicial object of $\operatorname{\mathcal{C}}$ satisfying $\overline{X}_{-1} = X$, and let $X_{\bullet } = \underline{X}_{\bullet } |_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) }$ denote its underlying semisimplicial object. We will say that $\overline{X}_{\bullet }$ exhibits $X$ as a geometric realization of $X_{\bullet }$ if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, in the sense of Variant 7.1.3.5. Similarly, if $\overline{X}^{\bullet }$ is an augmented cosemisimplicial object of $\operatorname{\mathcal{C}}$ satisfying $\overline{X}^{-1} = X$ and $X^{\bullet } = \overline{X}^{\bullet }|_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}) }$ is the underlying cosemisimplicial object, we say that $\overline{X}^{\bullet }$ exhibits $X$ as a totalization of $X^{\bullet }$ if it is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, in the sense of Definition 7.1.3.4.
Remark 10.2.2.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X_{\bullet }$ be a semisimplicial object of $\operatorname{\mathcal{C}}$, and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then $X$ is a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.2.2.9) if and only if there exists an augmented semisimplicial object $\overline{X}_{\bullet }$ which exhibits $X$ as a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.2.2.23). See Remark 7.1.3.7.
By virtue of Example 4.3.2.15, we can formulate Proposition 10.2.2.11 as follows:
Proposition 10.2.2.25. Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $X_{\bullet }$ be a semisimplicial object of $\operatorname{\mathcal{C}}$. The following data are equivalent:
Extensions of $X_{\bullet }$ to an augmented semisimplicial object $\overline{X}_{\bullet }$ satisfying $\overline{X}_{-1} = X$.
Morphisms $\epsilon : X_0 \rightarrow X$ satisfying $\epsilon \circ d^{1}_0 = \epsilon \circ d^{1}_1$, where $d^{1}_0, d^{1}_1: X_1 \rightrightarrows X_0$ are the face operators of the semisimplicial object $X_{\bullet }$.
Here the equivalence is implemented by taking $\epsilon $ to be the face operator $d^{0}_{0}: X_{0} \rightarrow X_{-1}$ of Remark 10.2.2.19.
Remark 10.2.2.26. In the situation of Proposition 10.2.2.25, the augmented semisimplicial object $\overline{X}_{\bullet }$ exhibits $X$ as a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.2.1.16) if and only if it the morphism $\epsilon $ exhibits $X$ as a coequalizer of the face operators $d^{1}_0, d^{1}_1: X_1 \rightrightarrows X_0$.
Combining Propositions 10.2.2.25 and 1.1.1.9, we obtain an explicit characterization of augmented semisimplicial objects of ordinary categories:
Corollary 10.2.2.27. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ X_ n \} _{n \geq -1}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then a system of morphisms $\{ d^{n}_{i}: X_{n} \rightarrow X_{n-1} \} _{0 \leq i \leq n}$ arise as the face operators of an augmented semisimplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy the following condition:
For all integers $n > 0$ and $0 \leq i < j \leq n$, we have an equality $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as morphisms from $X_{n}$ to $X_{n-2}$).
If this condition is satisfied, then the augmented semisimplicial object $X_{\bullet }$ is uniquely determined.
Variant 10.2.2.28. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ X_ n \} _{n \geq -1}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then morphisms are the face and degeneracy operators for an augmented simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy the following conditions:
For all integers $n > 0$ and $0 \leq i < j \leq n$, we have an equality $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as morphisms from $X_{n}$ to $X_{n-2}$).
For all integers $0 \leq i \leq j \leq n$, we have an equality $s^{n+1}_{i} \circ s^{n}_ j = s^{n+1}_{j+1} \circ s^{n}_ i$ (as morphisms from $X_{n}$ to $X_{n+2}$).
For all integers $0 \leq i,j \leq n$, we have an equality
(as morphisms from $X_{n}$ to $X_{n}$).
\[ \{ d^{n}_{i}: X_{n} \rightarrow X_{n-1} \} _{ 0 \leq i \leq n} \quad \quad \{ s^{n}_{i}: X_{n} \rightarrow X_{n+1} \} _{0 \leq i \leq n} \]
\[ d^{n+1}_{i} \circ s^{n}_ j = \begin{cases} s^{n-1}_{j-1} \circ d^{n}_ i & \text{ if } i < j \\ \operatorname{id}_{ X_ n } & \text{ if } i = j \text{ or } i = j + 1 \\ s^{n-1}_{j} \circ d^{n}_{i-1} & \text{ if } i > j+1 \end{cases} \]
If these conditions are satisfied, then the augmented simplicial object $X_{\bullet }$ is uniquely determined.
Proof. Combine Proposition 1.1.2.14, Remark 10.2.2.18, and Corollary 10.2.2.27. $\square$
We close this section with a few remarks concerning the relationship between simplicial and semisimplicial objects.
Proposition 10.2.2.29. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite coproducts and let $X_{\bullet }$ be a semisimplicial object of $\operatorname{\mathcal{C}}$. Then there exists a simplicial object $X_{\bullet }^{+}$ of $\operatorname{\mathcal{C}}$ and a natural transformation of semisimplicial objects $f_{\bullet }: X_{\bullet } \rightarrow X_{\bullet }^{+}$ which exhibits $X_{\bullet }^{+}$ as a left Kan extension of $X_{\bullet }$ along the inclusion map $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$.
Proof. We will show that $X_{\bullet }$ satisfies the criterion of Proposition 7.3.5.1. Fix an object $[n] \in \operatorname{{\bf \Delta }}$, let $\operatorname{\mathcal{E}}$ denote the fiber product $\operatorname{{\bf \Delta }}_{\operatorname{inj}} \times _{ \operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{ [n] / }$, and let $F$ denote the composite map
\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}^{\operatorname{op}}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}. \]
We wish to show that $F$ admits a colimit in $\operatorname{\mathcal{C}}$.
By definition, objects of the category $\operatorname{\mathcal{E}}$ can be identified with pairs $([m], \alpha )$, where $[m]$ is an object of $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ is an integer and $\alpha : [n] \rightarrow [m]$ is a nondecreasing function. Let $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ denote the full subcategory spanned by those objects $([m], \alpha )$ where $\alpha $ is a surjection. Note that any morphism $\alpha : [n] \rightarrow [m]$ in $\operatorname{{\bf \Delta }}$ factors uniquely as a composition $[n] \xrightarrow {\alpha '} [m'] \xrightarrow {\beta } [m]$, where $\alpha '$ is a surjection and $\beta $ is an injection. The pair $([m'], \alpha ')$ is then an object of the subcategory $\operatorname{\mathcal{E}}_0$, and the morphism $\beta : ([m'], \alpha ') \rightarrow ([m], \alpha )$ exhibits exhibits $([m'], \alpha ')$ as a $\operatorname{\mathcal{E}}_0$-coreflection of $([m], \alpha )$ (see Definition 6.2.2.1). It follows that the inclusion map $\operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}^{\operatorname{op}}_0 ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}^{\operatorname{op}} )$ is right cofinal (Corollary 7.2.2.7). Consequently, to show that $F$ admits a colimit in $\operatorname{\mathcal{C}}$, it will suffice to show that the restriction $F|_{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}^{\operatorname{op}}_{0} ) }$ admits a colimit in $\operatorname{\mathcal{C}}$ (Corollary 7.2.2.10). Since the category $\operatorname{\mathcal{E}}_0$ has finitely many objects and only identity morphisms, this follows from our assumption that $\operatorname{\mathcal{C}}$ admits finite coproducts. $\square$
Remark 10.2.2.30. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$, and suppose we are given a morphism $f_{\bullet }: X_{\bullet } \rightarrow Y_{\bullet }$ of semisimplicial objects of $\operatorname{\mathcal{C}}$. It follows from the proof of Proposition 10.2.2.29 that $f_{\bullet }$ exhibits $Y_{\bullet }$ as a left Kan extension of $X_{\bullet }$ along the inclusion map $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$ if and only if, for every integer $n \geq 0$, the following condition is satisfied:
Let $E$ be the collection of all surjections $\alpha : [n] \twoheadrightarrow [m]$ in the category $\operatorname{{\bf \Delta }}$. For each $\alpha \in E$, let $g_{\alpha }: X_{m} \rightarrow Y_{n}$ be a composition of $f_{m}: X_{m} \rightarrow Y_{m}$ with the morphism $\alpha ^{\ast }: Y_{m} \rightarrow Y_{n}$. Then the collection $\{ g_{\alpha } \} _{\alpha \in E}$ exhibit $Y_{n}$ as a coproduct of the collection of objects $\{ X_{m} \} _{ (\alpha : [n] \twoheadrightarrow [m]) \in E }$.
Compare with Construction 3.3.1.6.