### 4.5.7 Isofibrations of Simplicial Sets

We now use the characterization of Proposition 4.5.6.1 to generalize the notion of isofibration to arbitrary simplicial sets.

Definition 4.5.7.1. 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.

Example 4.5.7.4. 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 4.5.2.4) and therefore anodyne (Corollary 3.3.7.5), so that $q$ has the right lifting property with respect to $i$ (Remark 3.1.2.6).

Warning 4.5.7.8. 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 4.5.4.3). 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.

We have the following generalization of Proposition 4.4.5.1:

Proposition 4.5.7.10. 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 4.5.3.10, so the existence of the desired solution follows from our assumption that $q$ is an isofibration.
$\square$

Corollary 4.5.7.11. 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 4.5.7.10 in the special case $A = \emptyset $.
$\square$

Proposition 4.5.7.12. 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 4.5.3.10, so the existence of the desired solution follows from our assumption that $q$ is an isofibration.
$\square$

Corollary 4.5.7.13. 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 4.5.7.12 in the special case $S = \Delta ^0$.
$\square$

Proposition 4.5.7.14. 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 4.5.7.4 and a categorical equivalence by virtue of Proposition 4.5.2.9. Conversely, suppose that $q$ is both an isofibration and a categorical equivalence. Using Exercise 3.1.6.11, 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 4.5.2.9), so that $q'$ is also a categorical equivalence (Remark 4.5.2.6). 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.
$\square$