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10.1.4 The Čech Nerve

Let $S$ be a set. For each $n \geq 0$, we let $\operatorname{\check{C}}_{n}( S ) = \operatorname{Hom}( [n], S )$ denote the collection of functions from the set $[n] = \{ 0 < 1 < \cdots < n \} $ into $S$. The construction $[n] \mapsto \operatorname{\check{C}}_{n}(S)$ determines a simplicial set $\operatorname{\check{C}}_{\bullet }(S)$, which we will refer to as the Čech nerve of $S$. In this section, we study an $\infty $-categorical counterpart of this construction.

Definition 10.1.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. We will say that $C_{\bullet }$ is a Čech nerve if, for every integer $n \geq 0$, the following condition is satisfied:

$(\ast _ n)$

For $0 \leq i \leq n$, let $\nu _{i}: C_{n} \rightarrow C_{0}$ be the morphism of $\operatorname{\mathcal{C}}$ induced by the inclusion $[0] \simeq \{ i \} \subseteq [n]$. Then the morphisms $\{ \nu _{i} \} _{0 \leq i \leq n}$ exhibit $C_{n}$ as a product of $(n+1)$-copies of $C_0$.

Remark 10.1.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be a simplicial object of $\operatorname{\mathcal{C}}$. Then $C_{\bullet }$ is a Čech nerve if and only if it is right Kan extended from the full subcategory $\{ [0] \} \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$.

Definition 10.1.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We will say that a simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čech nerve of $X$ if $C_{\bullet }$ is a Čech nerve (in the sense of Definition 10.1.4.1) and $C_{0} = X$.

Notation 10.1.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. It follows from Remarks 10.1.4.2 and 7.3.6.6 that if $X$ admits a Čech nerve $C_{\bullet }$, then the simplicial object $C_{\bullet }$ is determined up to isomorphism and depends functorially on $X$. To emphasize this dependence, we will denote $C_{\bullet }$ by $\operatorname{\check{C}}_{\bullet }(X)$ and refer to it as the Čech nerve of $X$.

Proposition 10.1.4.5 (Existence). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then $X$ admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X)$ if and only if, for every nonempty finite set $J$, there exists a product of $J$ copies of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof. For every integer $n \geq 0$, the category $\{ [0] \} \times _{\operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{/ [n] }$ is isomorphic to the finite set $\{ 0, 1, \cdots , n \} $, regarded as a category having only identity morphisms. By virtue of Remark 10.1.4.2, the desired result is a special case of the existence criterion for Kan extensions (Corollary 7.3.5.8). $\square$

Corollary 10.1.4.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite products. Then every object $X \in \operatorname{\mathcal{C}}$ admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X)$.

Remark 10.1.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves finite products. Then the induced functor $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ), \operatorname{\mathcal{D}})$ carries Čech nerves to Čech nerves. In particular, if $X$ is an object of $\operatorname{\mathcal{C}}$ which admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X)$, then the image $Y = F(X)$ also admits a Čech nerve, given by $\operatorname{\check{C}}_{\bullet }(Y) = F( \operatorname{\check{C}}_{\bullet }(X) )$.

Corollary 10.1.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite products. Then the evaluation functor

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad C_{\bullet } \mapsto C_{0} \]

admits a right adjoint, given on objects by the Čech nerve $X \mapsto \operatorname{\check{C}}_{\bullet }(X)$.

We now study a relative version of Definition 10.1.4.1.

Definition 10.1.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, which we identify with a simplicial object $C'_{\bullet }$ of the $\infty $-category $\operatorname{\mathcal{C}}_{ / C_{-1} }$ (see Remark 10.1.1.15). We will say that $C_{\bullet }$ is a Čech nerve if the simplicial object $C'_{\bullet }$ is a Čech nerve in the $\infty $-category $\operatorname{\mathcal{C}}_{/C_{-1} }$, in the sense of Definition 10.1.4.1.

Remark 10.1.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Stated more informally, an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čech nerve if, for every integer $n \geq 0$, it exhibits $C_{n}$ as an iterated fiber product

\[ C_{0} \times _{ C_{-1} } C_{0} \times _{ C_{-1} } \cdots \times _{ C_{-1} } C_{0} \]

(where there are $(n+1)$ factors).

Remark 10.1.4.11. In the augmented simplex category $\operatorname{{\bf \Delta }}_{+}$, there is unique morphism $\delta ^{0}_{0}: [-1] \rightarrow [0]$. This morphism determines a fully faithful functor $[1] \rightarrow \operatorname{{\bf \Delta }}_{+}$, whose image is the full subcategory $\operatorname{\mathcal{J}}\subseteq \operatorname{{\bf \Delta }}_{+}$ spanned by the objects $[0]$ and $[-1]$. Combining Remarks 10.1.4.2 and 7.3.2.4, we see that an augmented simplicial object $C_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$ is a Čech nerve if and only if it is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{\mathcal{J}}^{\operatorname{op}} ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} )$.

Definition 10.1.4.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čech nerve of $f$ if $C_{\bullet }$ is a Čech nerve (in the sense of Definition 10.1.4.9) and the face operator $d^{0}_{0}: C_{0} \rightarrow C_{-1}$ coincides with the morphism $f$.

Notation 10.1.4.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. It follows from Remarks 10.1.4.11 and 7.3.6.6 that if $f$ admits a Čech nerve $C_{\bullet }$, then the augmented simplicial object $C_{\bullet }$ is determined up to isomorphism and depends functorially on $f$. To emphasize this dependence, we will denote $C_{\bullet }$ by $\operatorname{\check{C}}_{\bullet }(X/Y)$ and refer to it as the Čech nerve of the morphism $f: X \rightarrow Y$. Alternatively, we can identify $\operatorname{\check{C}}_{\bullet }(X/Y)$ with the simplicial object of $\operatorname{\mathcal{C}}_{/Y}$ given by the Čech nerve of $f$ (in the sense of Notation 10.1.4.4).

Proposition 10.1.4.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y)$.

Proof. Apply Corollary 10.1.4.6 to the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, which admits finite products by virtue of our assumption that $\operatorname{\mathcal{C}}$ admits pullbacks (Corollary 7.6.3.20). $\square$

Remark 10.1.4.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves fiber products. Then the induced functor of augmented simplicial objects $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ), \operatorname{\mathcal{D}})$ carries Čech nerves to Čech nerves (see Remark 10.1.4.7). In particular, if $u: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ which admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y)$, then the morphism $F(u): F(X) \rightarrow F(Y)$ admits a Čech nerve in the $\infty $-category $\operatorname{\mathcal{D}}$, given by $F( \operatorname{\check{C}}_{\bullet }(X/Y) )$.

Corollary 10.1.4.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then the forgetful functor

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \quad \quad C_{\bullet } \mapsto (d^{0}_{0}: C_0 \rightarrow C_{-1}) \]

admits a right adjoint, given on objects by the construction $(f: X \rightarrow Y) \mapsto \operatorname{\check{C}}_{\bullet }(X/Y)$.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits fiber products, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Our next goal is to show that splittings of the Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y)$ (in the sense of Definition 10.1.3.3) can be identified with sections of $f$ (Proposition 10.1.4.20). We begin with some general remarks.

Notation 10.1.4.17. 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 $\rho : \operatorname{Ret}\rightarrow \operatorname{{\bf \Delta }}$ which carries $i$ to the inclusion map $[0] \hookrightarrow [1]$ and $r$ to the constant function $[1] \twoheadrightarrow [0]$. Note that $\rho $ factors through the subcategory $\operatorname{{\bf \Delta }}_{ \mathrm{min} }$ introduced in Notation 10.1.3.1. Moreover, it $\rho $ induces an isomorphism from $\operatorname{Ret}$ to the full subcategory of $\operatorname{{\bf \Delta }}_{ \mathrm{min}}$ spanned by the objects $[0] = \rho (Y)$ and $[1] = \rho (X)$. In what follows, we will abuse notation by identifying $\operatorname{Ret}$ with the image of functor $\rho $.

In what follows, we let $C_{+}: \operatorname{{\bf \Delta }}_{+} \rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$ denote the concatenation functor of Notation 10.1.4.17, given on objects by the formula $C_{+}( [m] ) = [m+1] \simeq [0] \star [m]$. Note that the functor $\rho $ of Notation 10.1.4.17 fits into a commutative diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{{\bf \Delta }}_{\mathrm{min}} & \operatorname{{\bf \Delta }}_{+} \ar [l]_{ C_{+} } \\ \operatorname{Ret}\ar [u]^{\rho } & [1], \ar [l] \ar [u] } \]

where the lower horizontal functor classifies the morphism $r: X \rightarrow Y$ of $\operatorname{Ret}$, and the right vertical functor classifies the morphism $\delta ^{0}_{0}: [-1] \rightarrow [0]$ in $\operatorname{{\bf \Delta }}_{+}$. For any $\infty $-category $\operatorname{\mathcal{C}}$, we obtain a commutative diagram of restriction functors

10.18
\begin{equation} \begin{gathered}\label{equation:retraction-diagram-from-splitting} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\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{Ret}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}). } \end{gathered} \end{equation}

Here the right vertical map carries an augmented simplicial object $Z_{\bullet }$ of $\operatorname{\mathcal{C}}$ to the augmentation $d^{0}_{0}: Z_0 \rightarrow Z_{-1}$, and the left vertical map carries any splitting of $Z_{\bullet }$ to a retraction diagram

\[ \xymatrix@R =50pt@C=50pt{ & Z_{0} \ar [dr]^{ d^{0}_{0} } & \\ Z_{-1} \ar [ur]^{h_{-1}} \ar [rr]^{\operatorname{id}} & & Z_{-1} } \]

where $h_{-1}$ is the “extra degeneracy” operator appearing in Remark 10.1.3.4.

Lemma 10.1.4.18. 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 10.1.3.3). The following conditions are equivalent:

$(1)$

The functor $\overline{X}$ is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}} ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\mathrm{min}} )$ (see Notation 10.1.4.17).

$(2)$

The augmented simplicial object $X_{\bullet }$ is a Čech nerve (in the sense of Definition 10.1.4.9).

Proof. Let $\operatorname{\mathcal{J}}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{+}$ spanned by the objects $[-1]$ and $[0]$. By virtue of Remark 10.1.4.11, the augmented simplicial object $X_{\bullet }$ is a Čech nerve if and only if it is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}^{\operatorname{op}} )$. Fix an integer $n \geq 0$. We will complete the proof by showing that the following conditions are equivalent:

$(1_ n)$

The functor $\overline{X}$ is right Kan extended from $\operatorname{N}_{\bullet }(\operatorname{Ret}^{\operatorname{op}})$ at the object $[n] \in \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\mathrm{min}} )$.

$(2_ n)$

The augmented simplicial object $X_{\bullet }$ is right Kan extended from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}^{\operatorname{op}})$ at the object $[n-1] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+}$.

Let $\operatorname{Ret}_{ / [n] }$ denote the fiber product $\operatorname{Ret}\times _{ \operatorname{{\bf \Delta }}_{\mathrm{min}} } ( \operatorname{{\bf \Delta }}_{ \mathrm{min} } )_{ / [n] }$, and define $\operatorname{\mathcal{J}}_{ / [n-1] }$ similarly. Condition $(1_ n)$ asserts that the composition

\[ T: \operatorname{N}_{\bullet }( \operatorname{Ret}_{ / [n] } )^{\triangleright } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}} )^{\triangleright }_{ / [n]} \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}} ) \xrightarrow {\overline{X}^{\operatorname{op}}} \operatorname{\mathcal{C}}^{\operatorname{op}} \]

is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Similarly, $(2_ n)$ asserts that the composition

\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{J}}_{ / [n-1] } )^{\triangleright } \xrightarrow {G^{\triangleright }} \operatorname{N}_{\bullet }( \operatorname{Ret}_{ / [n] } )^{\triangleright } \xrightarrow {T} \operatorname{\mathcal{C}}^{\operatorname{op}} \]

is a colimit diagram, where $G: \operatorname{N}_{\bullet }( \operatorname{\mathcal{J}}_{ / [n-1] } ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Ret}_{ / [n] } )$ is induced by the concatenation functor $C_+$. The functor $G$ admits a left adjoint $F$, which carries a morphism $\alpha : [m] \rightarrow [n]$ of $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ to the nondecreasing function

\[ \{ k \in [m]: \alpha (k) > 0 \} \xrightarrow {\alpha } \{ 1 < 2 < \cdots < n \} \simeq [n-1]. \]

In particular, the functor $G$ is right cofinal (Corollary 7.2.3.7), so the equivalence of $(1_ n)$ with $(2_ n)$ is a special case of Corollary 7.2.2.3). $\square$

Variant 10.1.4.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a functor $F: \operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$, so that $F$ determines a retraction diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{\operatorname{id}} & & Y } \]

in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The functor $F$ admits a right Kan extension $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\mathrm{min}} ) \rightarrow \operatorname{\mathcal{C}}$.

$(2)$

The morphism $r: X \rightarrow Y$ admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y)$.

Proof. We maintain the notations from the proof of Lemma 10.1.4.18. By virtue of Corollary 7.3.5.8, it will suffice to show that for every integer $n \geq 0$, the following conditions are equivalent:

$(1_ n)$

The diagram

\[ T_0: \operatorname{N}_{\bullet }( \operatorname{Ret}_{/[n]}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Ret}) \xrightarrow { F^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} \]

has a colimit in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

$(2_ n)$

The composition $T_0 \circ G$ has a colimit in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, where $G: \operatorname{N}_{\bullet }( \operatorname{\mathcal{J}}_{ / [n-1] } ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Ret}_{ / [n] } )$ is induced by the concatenation functor $[m] \mapsto [0] \star [m]$.

As in the proof of Lemma 10.1.4.18, the functor $G$ is right cofinal, so the equivalence of $(1_ n)$ and $(2_ n)$ is a special case of Corollary 7.2.2.10. $\square$

Proposition 10.1.4.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ spanned by those augmented simplicial objects of $\operatorname{\mathcal{C}}$ which are Čech nerves, and let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ denote its inverse image. Then the diagram (10.18) induces a trivial Kan fibration

\[ \theta : \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}). \]

Proof. Let $\operatorname{Fun}'( \Delta ^1, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by those morphisms $f: X \rightarrow Y$ which admit a Čech nerve, and let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}} ), \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Ret}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ spanned by those functors which admit a right Kan extension to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\mathrm{min} })$. We then have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}'( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}'( \Delta ^1, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}). } \]

Combining Lemma 10.1.4.18 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 10.1.4.19 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 4.5.2.23). 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 4.5.2.22). By construction, $\theta $ is a pullback of the restriction functor

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}), \]

which is an isofibration by virtue of Corollary 4.4.5.3. It follows that $\theta $ is both an isofibration and a categorical equivalence, and therefore a trivial Kan fibration (Proposition 4.5.5.20). $\square$

Remark 10.1.4.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. It follows from Proposition 10.1.4.20 that the inclusion map $\operatorname{Ret}\hookrightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$ induces a trivial Kan fibration

\[ \{ \operatorname{\check{C}}_{\bullet }(X/Y) \} \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) } \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \{ f \} \times _{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Ret}^{\operatorname{op}}), \operatorname{\mathcal{C}}). \]

Stated more informally, the datum of a splitting of the Čech nerve $\operatorname{\check{C}}_{\bullet }( X/ Y )$ is essentially equivalent to the datum of a morphism $i: Y \rightarrow X$ together with a homotopy from $f \circ i$ to the identity morphism $\operatorname{id}_{Y}$ (see Corollary 8.5.1.27).

Remark 10.1.4.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits fiber products. Then Proposition 10.1.4.14 guarantees that every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ admits a Čech nerve. In this case, the proof of Proposition 10.1.4.20 shows that the restriction functor $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Corollary 10.1.4.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. Then $\operatorname{\check{C}}_{\bullet }(X/Y)$ splits if and only if $f$ admits a right homotopy inverse.

Corollary 10.1.4.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. If $f$ admits a section, 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.