Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
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4.5.6 Lifting Property of Isofibrations

We now characterize isofibrations between $\infty $-categories by means of a lifting property.

Proposition 4.5.6.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. Then $F$ is an isofibration if and only if it satisfies the following condition:

$(\ast )$

Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset for which the inclusion $A \hookrightarrow B$ is a categorical equivalence. Then every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]

admits a solution.

We begin by proving a weak form of Proposition 4.5.6.1.

Lemma 4.5.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset with the property that the inclusion $A \hookrightarrow B$ is a categorical equivalence. Then every diagram $f_0: A \rightarrow \operatorname{\mathcal{C}}$ can be extended to a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$.

Proof. By virtue of Corollary 4.4.5.4, the restriction map $\theta : \operatorname{Fun}(B,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})^{\simeq }$ is a Kan fibration. Since the inclusion $A \hookrightarrow B$ is a categorical equivalence, the map $\theta $ is a homotopy equivalence of Kan complexes (Proposition 4.5.2.8). Invoking Proposition 3.3.7.4, we conclude that $\theta $ is a trivial Kan fibration. In particular, it is surjective on vertices. $\square$

Lemma 4.5.6.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset, and suppose we are given a pair of diagrams $f,g: B \rightarrow \operatorname{\mathcal{C}}$ together with a natural transformation $u_0: f|_{A} \rightarrow f'|_{A}$. If the inclusion $A \hookrightarrow B$ is a categorical equivalence, then $u_0$ can be lifted to a natural transformation $u: f \rightarrow f'$. Moreover, if $u_0$ is a natural isomorphism, then $u$ is automatically a natural isomorphism.

Proof. The existence of the natural isomorphism $u$ follows by applying Lemma 4.5.6.2 to the inclusion of simplicial sets

\[ (\Delta ^1 \times A) \coprod _{ (\operatorname{\partial \Delta }^1 \times A)} (\operatorname{\partial \Delta }^1 \times B) \hookrightarrow \Delta ^1 \times B, \]

which is a categorical equivalence by virtue of Corollary 4.5.3.10. We will complete the proof by showing that if $u_0$ is a natural isomorphism, then $u$ is a natural isomorphism.

Let us identify $u$ with a morphism of simplicial sets $v: B \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$, and let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$. Since $u_0$ is a natural isomorphism, then restriction $v|_{A}$ factors through the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}})$. Invoking Lemma 4.5.6.2, we conclude that $v|_{A}$ extends to a diagram $v': B \rightarrow \operatorname{Isom}(\operatorname{\mathcal{C}})$. Since the inclusion $A \hookrightarrow B$ is a categorical equivalence, the equality $v|_{A} = v'|_{A}$ guarantees that $v$ and $v'$ are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(B, \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) )$. Since the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ is replete (Example 4.4.1.13), we conclude that $v$ also factors through $\operatorname{Isom}(\operatorname{\mathcal{C}})$, so that $u$ is a natural isomorphism by virtue of Theorem 4.4.4.4. $\square$

Proof of Proposition 4.5.6.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume first that $F$ satisfies condition $(\ast )$ of Proposition 4.5.6.1; we will prove that $F$ is an isofibration. For $0 < i < n$, the inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a categorical equivalence (Corollary 4.5.2.11), so condition $(\ast )$ guarantees that $F$ is an inner fibration. Fix an object $C \in \operatorname{\mathcal{C}}$ and an isomorphism $u: D \rightarrow F(C)$ in the $\infty $-category $\operatorname{\mathcal{D}}$; we wish to show that $u$ can be lifted to an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$. By virtue of Corollary 4.4.3.9, we can assume that $u = G(v)$ for some functor $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{E}}$ is a contractible Kan complex and $v: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{E}}$. Since the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence (Example 4.5.1.13), condition $(\ast )$ guarantees the existence of a solution to the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \{ Y\} \ar [r]^-{ Y \mapsto C} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{E}}\ar [r]^-{G} \ar@ {-->}[ur]^{\overline{G}} & \operatorname{\mathcal{D}}. } \]

Then $\overline{u} = \overline{G}(v)$ is an isomorphism of $\operatorname{\mathcal{C}}$ having the desired property.

Now suppose that the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an isofibration; we wish to show that condition $(\ast )$ is satisfied. Let $B$ be a simplicial set and $A \subseteq B$ a simplicial subset for which the inclusion $A \hookrightarrow B$ is a categorical equivalence. We wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r]^-{f_0} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar [r]^-{\overline{f}} \ar@ {-->}[ur]^{f} & \operatorname{\mathcal{D}}} \]

admits a solution. Invoking Lemma 4.5.6.2, we see that $f_0$ can be extended to a morphism of simplicial sets $f': B \rightarrow \operatorname{\mathcal{C}}$. Let $\overline{f}'$ denote the composition $B \xrightarrow {f'} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$, so that $\overline{f}|_{A} = \overline{f}'|_{A}$. Invoking Lemma 4.5.6.3, we conclude that there exists an isomorphism $\overline{u}: \overline{f} \rightarrow \overline{f}'$ in the diagram $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{D}})$ whose image in $\operatorname{Fun}(A, \operatorname{\mathcal{D}})$ is the identity transformation $\operatorname{id}_{ \overline{f}|_{A} }$. Applying Proposition 4.4.5.6, we deduce that $\overline{u}$ can be lifted to an isomorphism $u: f \rightarrow f'$ in the diagram $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ whose image in $\operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is the identity transformation $\operatorname{id}_{ f_0 }$. The diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ then satisfies $f|_{A} = f_0$ and $F \circ f = \overline{f}$, as desired. $\square$