We now introduce a tool which is often useful for computing geometric realizations of simplicial objects.
Notation 10.2.6.1. We define a category $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ as follows:
The objects of $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ are linearly ordered sets $[n] = \{ 0 < 1 < \cdots < n \} $, where $n$ is a nonnegative integer.
A morphism from $[m]$ to $[n]$ in the category $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ is a nondecreasing function $\alpha : [m] \rightarrow [n]$ satisfying $\alpha (0) = 0$.
Remark 10.2.6.2. By construction, the category $\operatorname{{\bf \Delta }}_{ \mathrm{min}}$ is a (non-full) subcategory of the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.0.2. It can therefore also be regarded as a subcategory of the augmented simplex category $\operatorname{{\bf \Delta }}_{+}$ of Definition 10.2.1.10. The inclusion functor $\operatorname{{\bf \Delta }}_{\mathrm{min}} \hookrightarrow \operatorname{{\bf \Delta }}_{+}$ admits a left adjoint $C_{+}: \operatorname{{\bf \Delta }}_{+} \rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$, given concretely by the construction $C_{+}( [n] ) = [0] \star [n] \simeq [n+1]$. We will refer to $C_{+}$ as the concatenation functor. We let $C: \operatorname{{\bf \Delta }}\rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$ denote the restriction of $C_{+}$ to the simplex category $\operatorname{{\bf \Delta }}$, which we will also refer to as the concatenation functor.
Definition 10.2.6.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$ (Definition 10.2.1.12). A splitting of $X_{\bullet }$ is a functor $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ for which the composition is equal to $X_{\bullet }$; here $C_{+}$ denotes the concatenation functor $[n] \mapsto [n] \star [0]$ of Remark 10.2.6.2. We will say that the augmented simplicial object $X_{\bullet }$ is split if there exists a splitting of $X_{\bullet }$.
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \xrightarrow {C^{\operatorname{op}}_{+}} \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \xrightarrow { \overline{X} } \operatorname{\mathcal{C}} \]
Remark 10.2.6.4 (Extra Degeneracies). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. For every integer $n \geq -1$, the function belongs to the subcategory $\operatorname{{\bf \Delta }}_{\min } \subseteq \operatorname{{\bf \Delta }}$. If $\overline{X}$ is a splitting of $X_{\bullet }$, then evaluation on $\sigma ^{0}_{n+1}$ determines a morphism Heuristically, one can think of the morphisms $\{ h_{n} \} _{n \geq -1}$ as “extra” degeneracy operators on the augmented simplicial object $X_{\bullet }$. In the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, these operators satisfy the identities
\[ \sigma ^{0}_{n+1}: [n+2] \rightarrow [n+1] \quad \quad i \mapsto \begin{cases} 0 & \text{ if $i=0$ } \\ i-1 & \text{ if $i > 0$ } \end{cases} \]
\[ h_ n: X_{n} = \overline{X}( [n+1] ) \rightarrow \overline{X}( [n+2] ) = X_{n+1}. \]
10.11
\begin{eqnarray} \label{equation:face-of-extra-degeneracy1} d^{n+1}_{i} \circ h_ n & \sim & \begin{cases} \operatorname{id}_{ X_{n} } & \text{ if $i = 0$ } \\ h_{n-1} \circ d^{n}_{i-1} & \text{ otherwise } \end{cases}\end{eqnarray}
10.14
\begin{eqnarray} \label{equation:face-of-extra-degeneracy2} s^{n+1}_{i} \circ h_{n} & \sim & \begin{cases} h_{n+1} \circ h_{n} & \text{ if $i=0$ } \\ h_{n+1} \circ s^{n}_{i-1} & \text{ otherwise. } \end{cases}\end{eqnarray}
Exercise 10.2.6.5. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. Show that the construction of Remark 10.2.6.4 determines a bijection from the set of splittings of $X_{\bullet }$ (in the sense of Definition 10.2.6.3) to the collection of systems $\{ h_{n}: X_{n} \rightarrow X_{n+1} \} _{n \geq -1}$ satisfying the identities (10.11) and (10.14).
Example 10.2.6.6. Let $A_{\bullet }$ be an augmented simplicial abelian group, and let denote its augmented Moore complex (Remark 10.2.2.21). Suppose we are given a splitting of $A_{\bullet }$, and let $\{ h_{n}: A_{n} \rightarrow A_{n+1} \} _{n \geq -1}$ be the extra degeneracy operators described in Remark 10.2.6.4. Then the collection $\{ h_{n} \} $ is a contracting homotopy for $\mathrm{C}_{\ast }^{\operatorname{aug}}( A )$, in the sense of Definition 2.5.0.5: that is, the homomorphism is equal to the identity for each $n \geq -1$ (where we adopt the convention that $h_{n} \circ \partial = 0$ for $n = -1$). This follows from the calculation where the final equality follows from the identities (10.11).
\[ \mathrm{C}_{\ast }^{\mathrm{aug}}( A ) = ( \cdots \rightarrow A_{2} \xrightarrow {\partial } A_{1} \xrightarrow {\partial } A_0 \xrightarrow {\partial } A_{-1} ) \]
\[ ( h_{n-1} \circ \partial + \partial \circ h_{n} ): A_{n} \rightarrow A_{n} \]
\begin{eqnarray*} h_{n-1} \circ \partial + \partial \circ h_{n} & = & (\sum _{i=0}^{n} (-1)^{i} h_{n-1} \circ d^{n}_{i}) + (\sum _{j=0}^{n+1} (-1)^{j} d^{n+1}_{j} \circ h_{n}) \\ & = & (\sum _{i=0}^{n} (-1)^{i} (h_{n-1} \circ d^{n}_{i} - d^{n+1}_{i+1} \circ h_{n} ) ) + d^{n+1}_{0} \circ h_{n} \\ & = & \operatorname{id}_{ A_{n} }. \end{eqnarray*}
Variant 10.2.6.7. In the situation of Example 10.2.6.6, let $\mathrm{N}_{\ast }^{\mathrm{aug}}(A)$ denote the augmented normalized Moore complex of $A_{\bullet }$ (Variant 10.2.2.22). It follows from (10.14) that, for every integer $n \geq 0$, the operator carries degenerate $n$-simplices of $A_{\bullet }$ to degenerate $(n+1)$-simplices of $A_{\bullet }$, and therefore descends to an operator $\overline{h}_{n}: \mathrm{N}^{\operatorname{aug}}_{n}(A) \rightarrow \mathrm{N}^{\operatorname{aug}}_{n+1}(A)$. The collection of homomorphisms $\{ \overline{h}_{n} \} $ then determine a contracting homotopy for the chain complex $\mathrm{N}_{\ast }^{\mathrm{aug}}(A)$.
\[ \mathrm{C}^{\operatorname{aug}}_{n}(A) = A_{n} \xrightarrow {h_ n} A_{n+1} = \mathrm{C}^{\operatorname{aug}}_{n+1}(A) \]
Warning 10.2.6.8. Let $A_{\bullet }$ be an augmented simplicial abelian group. In general, not every contracting homotopy for the chain complex $\mathrm{N}_{\ast }^{\mathrm{aug}}(A)$ can be obtained from the construction of Variant 10.2.6.7. A splitting of $A_{\bullet }$ determines a system of homomorphisms $\{ h_{n}: A_{n} \rightarrow A_{n+1} \} _{n \geq 0}$ which satisfy the identity $h_{n+1} \circ h_{n} = s^{n+1}_{0} \circ h_{n}$ (Remark 10.2.6.4). In particular, the composition $h_{n+1} \circ h_{n}$ carries every $n$-simplex of $A_{\bullet }$ to a degenerate $(n+2)$-simplex of $A_{\bullet }$. It follows that the composite map vanishes; for a general contracting homotopy, the analogous statement need not be true.
\[ \mathrm{N}_{n}^{\operatorname{aug}}( A ) \xrightarrow { \overline{h}_{n} } \mathrm{N}_{n+1}^{\operatorname{aug}}( A ) \xrightarrow { \overline{h}_{n+1} } \mathrm{N}_{n+2}^{\operatorname{aug}}(A) \]
The utility of Definition 10.2.6.3 stems from the following:
Proposition 10.2.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. If $X_{\bullet }$ is split, then it is a colimit diagram in $\operatorname{\mathcal{C}}$.
Proof. Let $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \mathrm{min} }^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be a splitting of $X_{\bullet }$, and let $C_{+}: \operatorname{{\bf \Delta }}_{+} \rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$ denote the concatenation functor of Remark 10.2.6.2. Let us abuse notation by identifying $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} )$ with the cone $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )^{\triangleright }$. We wish to show that the augmented simplicial object
is a colimit diagram in $\operatorname{\mathcal{C}}$.
Note that $[0]$ is initial when viewed as an object of the category $\operatorname{{\bf \Delta }}_{ \mathrm{min} }$, and therefore final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )$. Unwinding the definitions, we see that the functor $\operatorname{N}_{\bullet }( C_{+}^{\operatorname{op}} )$ factors as a composition
where $R$ is the identity when restricted to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )$ and carries the cone point of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )^{\triangleright }$ to $[0]$. Applying Corollary 7.2.2.6, we deduce that $(\overline{X} \circ R): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram. Consequently, to show that $X_{\bullet }$ is a colimit diagram, it will suffice to show that the functor $\operatorname{N}_{\bullet }( C^{\operatorname{op}} ): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )$ is right cofinal (Corollary 7.2.2.3). This is a special case of Corollary 7.2.3.7, since the concatenation functor $C$ is left adjoint to the inclusion $\operatorname{{\bf \Delta }}_{\mathrm{min}} \hookrightarrow \operatorname{{\bf \Delta }}$ (Remark 10.2.6.2). $\square$
Remark 10.2.6.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, so that $F(X_{\bullet } )$ is an augmented cosimplicial object of $\operatorname{\mathcal{D}}$. Composition with the functor $F$ carries splittings of $X_{\bullet }$ to splittings of $F( X_{\bullet } )$. Consequently, if $X_{\bullet }$ is split, then $F( X_{\bullet } )$ is also split. In particular, if $X_{\bullet }$ is split, then $F( X_{\bullet } )$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Proposition 10.2.6.9).
Variant 10.2.6.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. A splitting of $X_{\bullet }$ is a functor $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ for which the composition is equal to $X_{\bullet }$; here $C$ denotes the concatenation functor $[n] \mapsto [n] \star [0]$ of Remark 10.2.6.2. We will say that the simplicial object $X_{\bullet }$ is split if there exists a splitting of $X_{\bullet }$.
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \xrightarrow {C^{\operatorname{op}}} \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \xrightarrow { \overline{X} } \operatorname{\mathcal{C}} \]
Warning 10.2.6.12. The terminology of Variant 10.2.6.11 (and Definition 10.2.6.3) is potentially confusing. We will use the term split simplicial object to refer to a simplicial object $X_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$ for which there exists a splitting $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$. Unless otherwise specified, we do not assume that a particular splitting has been chosen. Beware that $\overline{X}$ is not uniquely determined by $X_{\bullet }$. However, the underlying augmented simplicial object is determined up to isomorphism by $X_{\bullet }$: by virtue of Proposition 10.2.6.9, it is an extension of $X_{\bullet }$ to a colimit diagram in $\operatorname{\mathcal{C}}$.
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \xrightarrow { \operatorname{N}_{\bullet }( C_{+}^{\operatorname{op}} ) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}} ) \xrightarrow { \overline{X} } \operatorname{\mathcal{C}} \]
Corollary 10.2.6.13. Let $X_{\bullet }$ be a split simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. Then $X_{\bullet }$ admits a geometric realization $| X_{\bullet } |$. Moreover, the geometric realization of $X_{\bullet }$ is preserved by any functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.
Proof. The first assertion follows from Proposition 10.2.6.9, and the second from Remark 10.2.6.10. $\square$
The Čechnerve construction of §10.2.5 provides an abundant supply of split simplicial objects.
Proposition 10.2.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. Then the augmented simplicial object $\operatorname{\check{C}}_{\bullet }(X/Y)$ splits if and only if $f$ admits a right homotopy inverse.
Corollary 10.2.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. If $f$ admits a right homotopy inverse, then $\operatorname{\check{C}}_{\bullet }(X/Y)$ is a colimit diagram: that is, it exhibits $Y$ as a geometric realization of its underlying simplicial object.
Proof. Combine Propositions 10.2.6.14 and 10.2.6.9. $\square$
Our proof of Proposition 10.2.6.14 will require some preliminaries.
Notation 10.2.6.16. Let $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ be the category introduced in Notation 10.2.6.1. For every integer $n$, we let $\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ spanned by the collection of objects $\{ [m] \} _{0 \leq m \leq n}$.
Example 10.2.6.17. For $n < 0$, the category $\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n}$ is empty. For $n = 0$, it contains a single object $[0]$, and no morphisms other than the identity morphism.
Example 10.2.6.18. Let $\operatorname{Ret}$ be the category introduced in Construction 8.5.0.2: that is, the category which is freely generated by a pair of morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ satisfying the identity $r \circ i= \operatorname{id}_{Y}$. By virtue of Exercise 8.5.0.3, there is a unique functor $\operatorname{Ret}\rightarrow \operatorname{{\bf \Delta }}$ which carries $i$ to the inclusion map $[0] \hookrightarrow [1]$ and $r$ to the constant function $[1] \twoheadrightarrow [0]$. This functor induces an isomorphism from $\operatorname{Ret}$ onto the subcategory $\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq 1} \subset \operatorname{{\bf \Delta }}$ of Notation 10.2.6.16.
Lemma 10.2.6.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, and let $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\mathrm{min}} ) \rightarrow \operatorname{\mathcal{C}}$ be a splitting of $X_{\bullet }$ (in the sense of Definition 10.2.6.3). For every integer $n$, the following conditions are equivalent:
The functor $\overline{X}$ is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )$ of Notation 10.2.6.16.
The augmented simplicial object $X_{\bullet }$ is $n$-coskeletal: that is, it is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq n})^{\operatorname{op}}$ (Definition 10.2.5.10).
Proof. For each integer $k \geq -1$, we will show that the following conditions are equivalent:
The functor $\overline{X}$ is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )^{\operatorname{op}}$ at the object $[k+1]$.
The functor $X_{\bullet }$ is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq n})^{\operatorname{op}}$ at the object $[k]$.
Let $(\operatorname{{\bf \Delta }}_{+}^{\leq n})_{ / [k] }$ denote the fiber product $(\operatorname{{\bf \Delta }}_{+})_{ / [k] } \times _{ \operatorname{{\bf \Delta }}_{+} } \operatorname{{\bf \Delta }}_{+}^{\leq n}$, and define $(\operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\leq n+1} )_{ / [k+1] }$ similarly. By virtue of Corollary 7.2.2.3, it will suffice to show that the concatentation functor $C_{+}$ induces a right cofinal functor
This follows from Corollary 7.2.3.7, since the functor $G$ admits a left adjoint $F$ (which carries a morphism $\alpha : [m] \rightarrow [k+1]$ of $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ to the nondecreasing function
Variant 10.2.6.20. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is equipped with a functor $T: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:
The functor $T$ admits a right Kan extension $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$.
The composite functor
can be extended to an $n$-coskeletal augmented simplicial object of $\operatorname{\mathcal{C}}$.
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq n})^{\operatorname{op}} \xrightarrow { C_{+} } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )^{\operatorname{op}} \xrightarrow {T} \operatorname{\mathcal{C}} \]
Proof. We maintain the notations from the proof of Lemma 10.2.6.19. By virtue of Corollary 7.3.5.8, it will suffice to show that for every integer $k \geq -1$, the following conditions are equivalent:
The diagram
admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.
The diagram
admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.
As in the proof of Lemma 10.2.6.19, the functor $G$ is right cofinal, so the equivalence of $(1_ k)$ and $(2_ k)$ is a special case of Corollary 7.2.2.10. $\square$
Proposition 10.2.6.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ be the full subcategory of $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ spanned by the $n$-coskeletal augmented simplicial objects of $\operatorname{\mathcal{C}}$, and let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ), \operatorname{\mathcal{C}})$ be its inverse image. Then precomposition with the concatenation functor $C_{+}$ induces a trivial Kan fibration
\[ \theta : \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n+1}_{\mathrm{min}})^{\operatorname{op}}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}}) } \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}). \]
Proof. Let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}})$ spanned by those functors which can be extended to $n$-coskeletal augmented simplicial objects of $\operatorname{\mathcal{C}}$, and define $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n+1}_{\mathrm{min}})^{\operatorname{op}}, \operatorname{\mathcal{C}})$ similarly. We then have a commutative diagram
Combining Lemma 10.2.6.19 with Corollary 7.3.6.15, we deduce that the upper vertical maps are trivial Kan fibrations; in particular, the upper half of the diagram is a categorical pullback square (Proposition 4.5.2.21). Variant 10.2.6.20 guarantees that the lower half of the square is a pullback diagram. Since bottom horizontal map is an isofibration of $\infty $-categories (Corollary 4.4.5.3), it is a categorical pullback square (Corollary 4.5.2.27). It follows that the outer rectangle is a categorical pullback square (Proposition 4.5.2.18), so that $\theta $ is an equivalence of $\infty $-categories (Proposition 4.5.2.26). Corollary 4.4.5.3 guarantees that $\theta $ is also an isofibration, so it is a trivial Kan fibration (Proposition 4.5.5.20). $\square$
Proof of Proposition 10.2.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. Applying Proposition 10.2.6.21 (in the case $n=0$), we see that precomposition with the inclusion map
of Example 10.2.6.18 induces a trivial Kan fibration
In particular, the left hand side is nonempty if and only if the right hand side is nonempty: that is, the Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$ splits if and only if $f$ has a right homotopy inverse. $\square$
We close this section by describing an important special class of split simplicial objects.
Construction 10.2.6.22 (Decalage). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. We let $\operatorname{Dec}_{+}( X )_{\bullet }$ denote the augmented simplicial object of $\operatorname{\mathcal{C}}$ given by the composition where $C_{+}$ denote the concatenation functor of Remark 10.2.6.2. We will refer to $\operatorname{Dec}_{+}(X)_{\bullet }$ as the augmented decalage of $X_{\bullet }$. We let $\operatorname{Dec}(X)_{\bullet }$ denote the underlying simplicial object of $\operatorname{Dec}_{+}(X)_{\bullet }$, given by the composition We will defer to $\operatorname{Dec}(X)_{\bullet }$ as the decalage of $\operatorname{Dec}(X)_{\bullet }$.
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \xrightarrow { \operatorname{N}_{\bullet }(C^{\operatorname{op}}_{+}) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}, \]
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { \operatorname{N}_{\bullet }(C^{\operatorname{op}}) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}. \]
Remark 10.2.6.23. More informally, the augmented decalage of a simplicial object $X_{\bullet }$ is given by the formula $\operatorname{Dec}_{+}(X)_{n} = X_{n+1}$. Moreover, for every pair of integers $0 \leq i \leq n$, the face and degeneracy operators coincide with the face and degeneracy operators
\[ d^{n}_{i}: \operatorname{Dec}_{+}(X)_{n} \rightarrow \operatorname{Dec}_{+}(X)_{n-1} \quad \quad s^{n}_{i}: \operatorname{Dec}_{+}(X)_{n} \rightarrow \operatorname{Dec}_{+}(X)_{n+1} \]
\[ d^{n+1}_{i+1}: X_{n+1} \rightarrow X_{n} \quad \quad s^{n+1}_{i+1}: X_{n+1} \rightarrow X_{n+2}. \]
Example 10.2.6.24. Let $X$ be a simplicial set. Then the decalage $\operatorname{Dec}(X)_{\bullet }$ can be identified with the disjoint union of coslice constructions $\coprod _{x} X_{x/}$, where the coproduct is indexed by the collection of all vertices $x \in X$.
Remark 10.2.6.25. Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. Then the augmented simplicial object $\operatorname{Dec}_{+}(X)_{\bullet }$ is split: it admits a splitting given by the diagram In particular, Proposition 10.2.6.9 guarantees that $\operatorname{Dec}_{+}(X)_{\bullet }$ is a colimit diagram in $\operatorname{\mathcal{C}}$: that is, it exhibits the object $X_{0} \in \operatorname{\mathcal{C}}$ as a geometric realization of the decalage $\operatorname{Dec}(X)_{\bullet }$.
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}. \]
Remark 10.2.6.26. Let $\iota : \operatorname{{\bf \Delta }}_{\mathrm{min}} \hookrightarrow \operatorname{{\bf \Delta }}$ denote the inclusion functor, and let denote the concatenation functor of Remark 10.2.6.2. There is a natural transformation $\eta : \operatorname{id}_{ \operatorname{{\bf \Delta }}} \rightarrow \iota \circ C$, which carries each object $[n] \in \operatorname{{\bf \Delta }}$ to the inclusion map If $X_{\bullet }$ is a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$, then composition with $\eta $ determines a natural transformation of simplicial objects $T_{\bullet }: \operatorname{Dec}(X)_{\bullet } \rightarrow X_{\bullet }$, given termwise by the face operator $\operatorname{Dec}(X)_{n} = X_{n+1} \xrightarrow { d^{n+1}_{0} } X_{n}$. The natural transformation $\eta $ is the unit of an adjunction between $\iota $ and $C$; it admits a compatible counit $\epsilon : C \circ \iota \rightarrow \operatorname{id}_{ \operatorname{{\bf \Delta }}_{ \mathrm{min}} }$, which carries each object $[n]$ to the quotient map We therefore have a commutative diagram in the functor category $\operatorname{Fun}( \operatorname{{\bf \Delta }}, \operatorname{{\bf \Delta }}_{\mathrm{min}} )$. If $\overline{X}$ is a splitting of the simplicial object $X_{\bullet }$, then precomposition with (10.17) determines a commutative diagram in the $\infty $-category of simplicial objects $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ), \operatorname{\mathcal{C}})$. Here $h_{\bullet }$ is given termwise by the extra degeneracy map $h_{n}: X_{n} \rightarrow X_{n+1} = \operatorname{Dec}(X)_{n}$ appearing in Remark 10.2.6.4. In particular, if $X_{\bullet }$ is a split simplicial object of $\operatorname{\mathcal{C}}$, then it is a retract of the decalage $\operatorname{Dec}(X)_{\bullet }$.
\[ C: \operatorname{{\bf \Delta }}\rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}} \quad \quad [n] \mapsto [0] \star [n] = [n+1] \]
\[ [n] \hookrightarrow [n+1] \quad \quad i \mapsto i+1. \]
\[ [n+1] \twoheadrightarrow [n] \quad \quad i \mapsto \mathrm{max}(0, i-1). \]
10.17
\begin{equation} \begin{gathered}\label{equation:comparison-map-with-decalage} \xymatrix@C =50pt@R=50pt{ & C \circ \iota \circ C \ar [dr]^{ \operatorname{id}_ C \circ \epsilon } & \\ C \ar [ur]^{ \eta \circ \operatorname{id}_{C} } \ar [rr]^{ \operatorname{id}_{C} } & & C } \end{gathered} \end{equation}
\[ \xymatrix@C =50pt@R=50pt{ & \operatorname{Dec}( X )_{\bullet } \ar [dr]^{ T_{\bullet } } & \\ X_{\bullet } \ar [ur]^{ h_{\bullet } } \ar [rr]^{ \operatorname{id}} & & X_{\bullet } } \]
Warning 10.2.6.27. Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. It follows from Remark 10.2.6.26 that every splitting of $X_{\bullet }$ determines a right homotopy inverse to the comparison map $T_{\bullet }: \operatorname{Dec}(X)_{\bullet } \rightarrow X_{\bullet }$. Beware that, in general, not every right homotopy inverse can be obtained in this way. For example, suppose that $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. Unwinding the definitions, we see that a morphism of simplicial objects from $X_{\bullet }$ to $\operatorname{Dec}(X)_{\bullet }$ is given by a collection of morphisms $h_{n}: X_{n} \rightarrow \operatorname{Dec}(X)_{n} = X_{n+1}$ which satisfy the identities for $0 < i \leq n+1$. Moreover, $h_{\bullet }$ is a right inverse of $T_{\bullet }$ if and only if it satisfies the further identity $d^{n+1}_{0} \circ h_ n = \operatorname{id}_{ X_ n }$ for each $n \geq 0$. However, $h_{\bullet }$ arises from a splitting of the simplicial object $X_{\bullet }$ only if it also satisfies the identities $s^{n+1}_{0} \circ h_{n} = h_{n+1} \circ h_{n}$; see Exercise 10.2.6.5 (compare with Warning 10.2.6.8).
\[ d^{n+1}_{i} \circ h_ n = h_{n-1} \circ d^{n}_{i-1} \quad \quad s^{n+1}_{i} \circ h_{n} = h_{n+1} \circ s^{n}_{i-1} \]