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4.5.5 Isofibrations of Simplicial Sets

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

Proposition 4.5.5.1. 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 4.5.5.1.

Lemma 4.5.5.2. 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 4.5.3.8). Invoking Proposition 3.3.7.4, we conclude that $\theta $ is a trivial Kan fibration. In particular, it is surjective on vertices. $\square$

Lemma 4.5.5.3. 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 $u: f \rightarrow g$. Moreover, if $u_0$ is a natural isomorphism, then $u$ is automatically a natural isomorphism.

Proof. The existence of the natural transformation $u$ follows by applying Lemma 4.5.5.2 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 4.5.4.15. 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, the restriction $v|_{A}$ factors through the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}})$. Invoking Lemma 4.5.5.2, 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 4.5.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume first that $F$ satisfies condition $(\ast )$ of Proposition 4.5.5.1; 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 4.5.3.14), 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 4.4.3.13, 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 4.5.5.2, 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 4.5.5.3, 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 Corollary 4.4.5.9, 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. $\square$

Proposition 4.5.5.1 has a converse:

Proposition 4.5.5.4. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then $i$ is a categorical equivalence if and only if the following condition is satisfied:

$(\ast )$

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories. Then every lifting problem

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

has a solution.

Proof. Assume that condition $(\ast )$ is satisfied; we will show that the morphism $i: A \hookrightarrow B$ is a categorical equivalence of simplicial sets (the converse follows from Proposition 4.5.5.1). Fix an $\infty $-category $\operatorname{\mathcal{E}}$; we wish to show that precomposition with $i$ induces a bijection $\theta : \pi _0( \operatorname{Fun}(B, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(A, \operatorname{\mathcal{E}})^{\simeq } )$. The surjectivity of $\theta $ follows by applying condition $(\ast )$ to the isofibration $\operatorname{\mathcal{E}}\rightarrow \Delta ^0$, and the injectivity of $\theta $ follows by applying $\theta $ to the isofibration $\operatorname{Isom}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}\times \operatorname{\mathcal{E}}$ of Corollary 4.4.5.5. $\square$

We now use the characterization of Proposition 4.5.5.1 to generalize the notion of isofibration to arbitrary simplicial sets.

Definition 4.5.5.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. We will say that $q$ is an isofibration 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

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution.

Remark 4.5.5.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. We have now given two a priori different definitions of an isofibration from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$:

  • According to Definition 4.4.1.4, an isofibration $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an inner fibration with the property that every isomorphism $u: D \rightarrow F(C)$ in the $\infty $-category $\operatorname{\mathcal{D}}$ can be lifted to an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$.

  • According to Definition 4.5.5.5, an isofibration $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a morphism of simplicial sets which has the right lifting property with respect to all monomorphisms $A \hookrightarrow B$ which are categorical equivalences.

However, these definitions are equivalent: this is the content of Proposition 4.4.5.1.

Remark 4.5.5.7. Let $q: X \rightarrow S$ be an isofibration of simplicial sets. Then $q$ is an inner fibration: that is, it has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^{n}$ for $0 < i < n$ (such inclusions are categorical equivalences, by virtue of Corollary 4.5.3.14). In particular, for each vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is an $\infty $-category (Remark 4.1.1.6). Moreover, if $S$ is an $\infty $-category, then $X$ is also an $\infty $-category (Remark 4.1.1.9).

Example 4.5.5.8. Let $q: X \rightarrow S$ be a Kan fibration of simplicial sets. Then $q$ is an isofibration. To prove this, we note that if a monomorphism of simplicial sets $i: A \hookrightarrow B$ is a categorical equivalence, then it is a weak homotopy equivalence (Remark 4.5.3.4) and therefore anodyne (Corollary 3.3.7.5), so that $q$ has the right lifting property with respect to $i$ (Remark 3.1.2.7).

Remark 4.5.5.9. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is an isofibration if and only if the opposite morphism $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ is an isofibration.

Remark 4.5.5.10. The collection of isofibrations is closed under retracts. That is, given a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X_{} \ar [r] \ar [d]^{q} & X'_{} \ar [d]^{q'} \ar [r] & X_{} \ar [d]^{q} \\ S_{} \ar [r] & S'_{} \ar [r] & S_{} } \]

where both horizontal compositions are the identity, if $q'$ is an isofibration, then so is $q$.

Remark 4.5.5.11. The collection of isofibrations is closed under pullback. That is, given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X'_{} \ar [d]^{q'} \ar [r] & X_{} \ar [d]^{q} \\ S'_{} \ar [r] & S_{} } \]

where $q$ is an isofibration, the morphism $q'$ is also an isofibration.

Warning 4.5.5.12. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X'_{} \ar [d]^{q'} \ar [r]^-{f'} & X_{} \ar [d]^{q} \\ S'_{} \ar [r]^-{f} & S_{}, } \]

where $q$ is an isofibration. If $f$ is an equivalence of $\infty $-categories, then $f'$ is also an equivalence of $\infty $-categories (Corollary 4.5.2.23). Beware that if $f$ is merely assumed to be a categorical equivalence of simplicial sets, then it is not necessarily true that $f'$ is a categorical equivalence of simplicial sets.

Remark 4.5.5.13. Let $p: X_{} \rightarrow Y_{}$ and $q: Y_{} \rightarrow Z_{}$ be isofibrations of simplicial sets. Then the composite map $(q \circ p): X_{} \rightarrow Z_{}$ is an isofibration of simplicial sets.

We have the following generalization of Proposition 4.4.5.1:

Proposition 4.5.5.14. Let $q: X \rightarrow S$ be an isofibration of simplicial sets and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then the restriction map

\[ q': \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) \]

is also an isofibration of simplicial sets.

Proof. 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. We wish to show that every lifting problem

\[ \xymatrix@C =50pt{ A' \ar [d] \ar [r] & \operatorname{Fun}(B,X) \ar [d]^{q'} \\ B' \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) } \]

admits a solution. Unwinding the definitions, we are reduced to the problem of solving an associated lifting problem

\[ \xymatrix@C =50pt{ (A \times B') \coprod _{ (A \times A') } (B \times A') \ar [d] \ar [r] & X \ar [d]^{q} \\ B \times B' \ar [r] \ar@ {-->}[ur] & S. } \]

The left vertical map in this diagram is a categorical equivalence by virtue of Corollary 4.5.4.15, so the existence of the desired solution follows from our assumption that $q$ is an isofibration. $\square$

Corollary 4.5.5.15. Let $q: X \rightarrow S$ be an isofibration of simplicial sets. For every simplicial set $B$, the induced map $\operatorname{Fun}(B, X) \rightarrow \operatorname{Fun}(B,S)$ is also an isofibration.

Proof. Apply Proposition 4.5.5.14 in the special case $A = \emptyset $. $\square$

Corollary 4.5.5.16. Let $q: X \rightarrow S$ be an isofibration of simplicial sets. Suppose we are given a morphism of simplicial sets $B \rightarrow S$ and a simplicial subset $A \subseteq B$. Then the restriction map $\theta : \operatorname{Fun}_{/S}( B, X ) \rightarrow \operatorname{Fun}_{/S}(A, X)$ is an isofibration of $\infty $-categories.

Proof. The morphism $\theta $ is a pullback of the isofibration $\operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S)$ of Proposition 4.5.5.14, and is therefore also an isofibration (Remark 4.5.5.11). We conclude by observing that since $q$ is an inner fibration (Remark 4.5.5.7), the simplicial sets $\operatorname{Fun}_{/S}(B,X)$ and $\operatorname{Fun}_{/S}(A,X)$ are $\infty $-categories (Proposition 4.1.4.6). $\square$

Remark 4.5.5.17. Suppose we are given a lifting problem in the category of simplicial sets

4.39
\begin{equation} \begin{gathered}\label{equation:typical-lifting-problem} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f_0} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{ \overline{f} } \ar@ {-->}[ur] & S, } \end{gathered} \end{equation}

where $q$ is an isofibration and $i$ is a monomorphism. It follows from Corollary 4.5.5.16 that, if we regard the morphisms $q$, $i$, and $\overline{f}$ as fixed, then the existence of a solution to the lifting problem (4.39) depends only on the isomorphism class of $f$ as an object of the $\infty $-category $\operatorname{Fun}_{/S}(A,X)$.

Proposition 4.5.5.18. Let $q: X \rightarrow S$ be an isofibration of simplicial sets and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. If $i$ is a categorical equivalence, then the restriction map

\[ q': \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) \]

is a trivial Kan fibration.

Proof. Let $B'$ be a simplicial set and let $A' \subseteq B'$ be a simplicial subset. We wish to show that every lifting problem

\[ \xymatrix@C =50pt{ A' \ar [d] \ar [r] & \operatorname{Fun}(B,X) \ar [d]^{q'} \\ B' \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) } \]

admits a solution. Unwinding the definitions, we are reduced to the problem of solving an associated lifting problem

\[ \xymatrix@C =50pt{ (A \times B') \coprod _{ (A \times A') } (B \times A') \ar [d] \ar [r] & X \ar [d]^{q} \\ B \times B' \ar [r] \ar@ {-->}[ur] & S. } \]

The left vertical map in this diagram is a categorical equivalence by virtue of Corollary 4.5.4.15, so the existence of the desired solution follows from our assumption that $q$ is an isofibration. $\square$

Corollary 4.5.5.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. If $i$ is a categorical equivalence, then the restriction functor $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is a trivial Kan fibration of simplicial sets.

Proof. Apply Proposition 4.5.5.18 in the special case $S = \Delta ^0$. $\square$

Proposition 4.5.5.20. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is a trivial Kan fibration if and only if it both an isofibration and a categorical equivalence.

Proof. If $q$ is a trivial Kan fibration, then it is an isofibration by virtue of Example 4.5.5.8 and a categorical equivalence by virtue of Proposition 4.5.3.11. Conversely, suppose that $q$ is both an isofibration and a categorical equivalence. Using Exercise 3.1.7.10, we can write $q$ as a composition $X \xrightarrow {q'} Y \xrightarrow {q''} S$, where $q'$ is a monomorphism and $q''$ is a trivial Kan fibration. Then $q''$ is a categorical equivalence (Proposition 4.5.3.11), so that $q'$ is also a categorical equivalence (Remark 4.5.3.5). Invoking our assumption that $q$ is an isofibration, we conclude that the lifting problem

\[ \xymatrix@C =50pt{ X \ar [r]^-{\operatorname{id}} \ar [d]^{q'} & X \ar [d]^{q} \\ Y \ar@ {-->}[ur]^{r} \ar [r]^-{q''} & S } \]

admits a solution. It follows that $q$ is a retract of the morphism $q''$, and is therefore also a trivial Kan fibration. $\square$