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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.

Remark 4.5.7.2. 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 4.5.7.1, 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 4.5.7.3. 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 4.5.2.11). 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 4.5.7.4. 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).

Remark 4.5.7.5. 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 4.5.7.6. 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 4.5.7.7. The collection 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 4.5.7.8. 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.

Remark 4.5.7.9. 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 4.5.7.10. 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. $\square$