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4.5.2 Categorical Pullback Squares

Recall that a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{q} \\ X_{1} \ar [r] & X } \]

is a homotopy pullback square if the induced map

\[ X_{01} \rightarrow X_0 \times _{X} X_{1} \hookrightarrow X_0 \times _{X}^{\mathrm{h}} X_1 \]

is a homotopy equivalence, where $X_0 \times _{X}^{\mathrm{h}} X_1$ is the homotopy fiber product of Construction 3.4.0.3 (see Corollary 3.4.1.6). In this section, we study an analogous condition in the setting of $\infty $-categories. We begin with a variant of Construction 3.4.0.3.

Construction 4.5.2.1 (The Homotopy Fiber Product of $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ denote the full subcategory spanned by the isomorphisms in $\operatorname{\mathcal{C}}$ (Example 4.4.1.13). If $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are $\infty $-categories equipped with functors $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$, we let $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ denote the iterated pullback

\[ \operatorname{\mathcal{C}}_0 \times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) } \operatorname{Isom}(\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}_1. \]

We will refer to $\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ as the homotopy fiber product of $\operatorname{\mathcal{C}}_0$ with $\operatorname{\mathcal{C}}_1$ over $\operatorname{\mathcal{C}}$. Note that the diagonal map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ induces a comparison map $\operatorname{\mathcal{C}}_{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1} \hookrightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1}$, which is a monomorphism of simplicial sets.

Remark 4.5.2.2. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. It follows from Corollary 4.4.5.5 that the projection map $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1$ is an isofibration. In particular, the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is an $\infty $-category. By construction, the objects of $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ can be identified with triples $(C_0, C_1, e)$, where $C_0$ is an object of $\operatorname{\mathcal{C}}_0$, $C_1$ is an object of $\operatorname{\mathcal{C}}$, and $e: F_0(C_0) \rightarrow F_1(C_1)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

Example 4.5.2.3. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. If $\operatorname{\mathcal{C}}$ is a Kan complex, then every morphism in $\operatorname{\mathcal{C}}$ is an isomorphism (Proposition 1.3.6.10): that is, we have $\operatorname{Isom}(\operatorname{\mathcal{C}}) = \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$. It follows that the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ of Construction 4.5.2.1 coincides with the homotopy fiber product introduced in Construction 3.4.0.3.

Remark 4.5.2.4. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Then there is a canonical isomorphism of simplicial sets

\[ (\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1)^{\operatorname{op}} \simeq \operatorname{\mathcal{C}}_{1}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_{0}^{\operatorname{op}}. \]

Remark 4.5.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a simplicial set. Using Theorem 4.4.4.4, we see that the natural identification $\operatorname{Fun}(X, \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \simeq \operatorname{Fun}( \Delta ^1, \operatorname{Fun}(X, \operatorname{\mathcal{C}}) )$ restricts to an isomorphism $\operatorname{Fun}(X, \operatorname{Isom}(\operatorname{\mathcal{C}}) ) \simeq \operatorname{Isom}( \operatorname{Fun}(X, \operatorname{\mathcal{C}}) )$. If $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ are functors of $\infty $-categories, we obtain a canonical isomorphism

\[ \operatorname{Fun}(X, \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1) \simeq \operatorname{Fun}(X, \operatorname{\mathcal{C}}_0) \times ^{\mathrm{h}}_{\operatorname{Fun}(X,\operatorname{\mathcal{C}})} \operatorname{Fun}(X, \operatorname{\mathcal{C}}_1). \]

Remark 4.5.2.6. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Applying Corollary 4.4.3.18 to the pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \ar [r] \ar [d] & \operatorname{Isom}(\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}, } \]

we deduce that the diagram of cores

\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1)^{\simeq } \ar [r] \ar [d] & \operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq } \ar [d] \\ \operatorname{\mathcal{C}}_0^{\simeq } \times \operatorname{\mathcal{C}}_{1}^{\simeq } \ar [r] & \operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq } } \]

is also a pullback square: that is, we have a canonical isomorphism of Kan complexes

\[ ( \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 )^{\simeq } \simeq \operatorname{\mathcal{C}}_0^{\simeq } \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}^{\simeq }} \operatorname{\mathcal{C}}_{1}^{\simeq }. \]

Definition 4.5.2.7. A commutative diagram of $\infty $-categories

4.21
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}

is a categorical pullback square if the composite map

\[ \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1} \hookrightarrow \operatorname{\mathcal{C}}_{0} \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1 \]

is an equivalence of $\infty $-categories.

Remark 4.5.2.8. Suppose we are given a categorical pullback diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \]

Then, for every simplicial set $K$, the induced diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X, \operatorname{\mathcal{C}}_{01} ) \ar [r] \ar [d] & \operatorname{Fun}(X, \operatorname{\mathcal{C}}_0) \ar [d] \\ \operatorname{Fun}(X, \operatorname{\mathcal{C}}_1) \ar [r] & \operatorname{Fun}( X, \operatorname{\mathcal{C}}) } \]

is also a categorical pullback square. This follows by combining Remarks 4.5.2.5 and 4.5.1.16.

Proposition 4.5.2.9. A commutative diagram of Kan complexes

4.22
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square5} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{q} \\ X_1 \ar [r] & X } \end{gathered} \end{equation}

is a categorical pullback square if and only if it is a homotopy pullback square.

In more general situations, the notions of homotopy pullback square and categorical pullback square are distinct:

Exercise 4.5.2.10. Show that the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \emptyset \ar [r] \ar [d] & \{ 0\} \ar [d] \\ \{ 1\} \ar [r] & \Delta ^1 } \]

is a categorical pullback square which is not a homotopy pullback square.

Exercise 4.5.2.11. Show that the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [r] \ar [d] & \Delta ^1 \ar [d] \\ \Delta ^1 \ar [r] & \Delta ^1 } \]

is a homotopy pullback square which is not a categorical pullback square.

Proposition 4.5.2.12. A commutative diagram of $\infty $-categories

4.23
\begin{equation} \begin{gathered}\label{equation:characterize-categorical-pullback2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

is a categorical pullback square if and only if, for every simplicial set $X$, the diagram of Kan complexes

4.24
\begin{equation} \begin{gathered}\label{equation:characterize-categorical-pullback22} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X, \operatorname{\mathcal{C}}_{01} )^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( X, \operatorname{\mathcal{C}}_0)^{\simeq } \ar [d] \\ \operatorname{Fun}(X, \operatorname{\mathcal{C}}_1)^{\simeq } \ar [r] & \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } } \end{gathered} \end{equation}

is a homotopy pullback square.

Proof. By definition, the diagram (4.23) is a categorical pullback square if and only if the induced map $\theta : \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is an equivalence of $\infty $-categories. Using the criterion of Proposition 4.5.1.22, we see that this is equivalent to the requirement that $\theta $ induces a homotopy equivalence $\theta _{X}: \operatorname{Fun}( X, \operatorname{\mathcal{C}}_{01} )^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1)^{\simeq }$ for every simplicial set $X$. Using Remarks 4.5.2.5 and 4.5.2.6, we can identify $\theta _{X}$ with the map

\[ \operatorname{Fun}( X, \operatorname{\mathcal{C}}_{01} )^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}}_0)^{\simeq } \times ^{\mathrm{h}}_{\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } } \operatorname{Fun}(X, \operatorname{\mathcal{C}}_1)^{\simeq } \]

determined by the commutative diagram (4.24). The desired result now follows from the criterion of Corollary 3.4.1.6. $\square$

Remark 4.5.2.13. In the situation of Proposition 4.5.2.12, it suffices to verify that the diagram (4.24) is a homotopy pullback square in the case where $X$ is an $\infty $-category. In fact, we will later see that it suffices to consider the case where $X = \Delta ^1$ (Corollary 4.5.7.4).

We now apply Proposition 4.5.2.12 to deduce some formal properties of the notion of categorical pullback square.

Proposition 4.5.2.14. A commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}} \]

is a categorical pullback square if and only if the induced diagram of opposite $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{\operatorname{op}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}^{\operatorname{op}}_0 \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_1 \ar [r] & \operatorname{\mathcal{C}}^{\operatorname{op}} } \]

is a categorical pullback square.

Proposition 4.5.2.15 (Symmetry). A commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}} \]

is a categorical pullback square if and only if the transposed diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_1 \ar [d] \\ \operatorname{\mathcal{C}}_0 \ar [r] & \operatorname{\mathcal{C}}} \]

is a categorical pullback square.

Proposition 4.5.2.16 (Transitivity). Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r] \ar [d] & \operatorname{\mathcal{C}}' \ar [d] \ar [r] & \operatorname{\mathcal{C}}'' \ar [d] \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}'', } \]

where the square on the the right is a categorical pullback. Then the square on the left is a categorical pullback if and only if the outer rectangle is a categorical pullback.

Proposition 4.5.2.17 (Homotopy Invariance). Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [dr]^{ F_{01} } \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}_{0} \ar [dd] \ar [dr]^{F_1} & \\ & \operatorname{\mathcal{D}}_{01} \ar [rr] \ar [dd] & & \operatorname{\mathcal{D}}_{0} \ar [dd] \\ \operatorname{\mathcal{C}}_{1} \ar [rr] \ar [dr]^{ F_1 } & & \operatorname{\mathcal{C}}\ar [dr]^{F} & \\ & \operatorname{\mathcal{D}}_{1} \ar [rr] & & \operatorname{\mathcal{D}}, } \]

where $F_0$, $F_1$, and $F$ are equivalences of $\infty $-categories. Then any two of the following conditions imply the third:

$(1)$

The back face

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}} \]

is a categorical pullback square.

$(2)$

The front face

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{D}}_0 \ar [d] \\ \operatorname{\mathcal{D}}_1 \ar [r] & \operatorname{\mathcal{D}}} \]

is a categorical pullback square.

$(3)$

The functor $F_{01}$ is an equivalence of $\infty $-categories.

Proof. Using Proposition 4.5.1.22, we see that $(3)$ is equivalent to the following:

$(3')$

For every simplicial set $X$, the functor $F_{01}$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}(X, \operatorname{\mathcal{C}}_{01} )^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}}_{01})^{\simeq }$.

The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3')$ now follow by combining Proposition 4.5.2.12 with Corollary 3.4.1.12. $\square$

Corollary 4.5.2.18. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] & \operatorname{\mathcal{C}}_1 \ar [l] \ar [d] \\ \operatorname{\mathcal{D}}_0 \ar [r] & \operatorname{\mathcal{D}}& \operatorname{\mathcal{D}}\ar [l] } \]

where the vertical maps are equivalences of $\infty $-categories. Then the induced map $\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$ is an equivalence of $\infty $-categories.

Proposition 4.5.2.19. Suppose we are given a commutative diagram of $\infty $-categories

4.25
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}

where $F$ is an equivalence of $\infty $-categories. Then (4.25) is a categorical pullback square if and only if $F'$ is an equivalence of $\infty $-categories.

Proposition 4.5.2.20. Suppose we are given a commutative diagram of $\infty $-categories

4.26
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square4} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \end{gathered} \end{equation}

where $U$ is an isofibration. Then (4.26) is a categorical pullback square if and only if the induced map $\theta : \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}'$ is an equivalence of $\infty $-categories.

Proof. For every simplicial set $X$, Corollary 4.4.5.7 guarantees that the induced map $\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq }$ is a Kan fibration. Combining Proposition 4.5.2.12 with Example 3.4.1.3, we see that (4.26) is a categorical pullback square if and only if it induces a homotopy equivalence

\[ \rho _{X}: \operatorname{Fun}( X, \operatorname{\mathcal{C}}')^{\simeq } \rightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}})^{\simeq } \times _{ \operatorname{Fun}( X, \operatorname{\mathcal{D}})^{\simeq } } \operatorname{Fun}(X, \operatorname{\mathcal{D}}')^{\simeq }, \]

for every simplicial set $X$. Using Corollary 4.4.3.18, we can identify $\rho _ X$ with the map $\operatorname{Fun}(X, \operatorname{\mathcal{C}}')^{\simeq } \rightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}' )^{\simeq }$ given by postcomposition with $\theta $. The desired result now follows from the criterion of Proposition 4.5.1.22. $\square$

Corollary 4.5.2.21. Suppose we are given a pullback diagram of $\infty $-categories

4.27
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square44} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}

If $U$ is an isofibration, then (4.27) is a categorical pullback square.

Corollary 4.5.2.22. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. If either $F_0$ or $F_1$ is an isofibration, then the comparison map

\[ \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \hookrightarrow \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \quad \quad (C_0, C_1) \mapsto (C_0, C_1, \operatorname{id}) \]

is an equivalence of $\infty $-categories.

Proof. This is a restatement of Corollary 4.5.2.21. $\square$

Corollary 4.5.2.23. Suppose we are given a pullback diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r]^-{F'} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{D}}' \ar [r]^-{F} & \operatorname{\mathcal{D}}, } \]

where $U$ is an isofibration. If $F$ is an equivalence of $\infty $-categories, then $F'$ is also an equivalence of $\infty $-categories.

Corollary 4.5.2.24. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix { \operatorname{\mathcal{C}}_0 \ar [r]^{U} \ar [d] & \operatorname{\mathcal{C}}\ar [d] & \operatorname{\mathcal{C}}_1 \ar [l] \ar [d] \\ \operatorname{\mathcal{D}}_0 \ar [r]^{V} & \operatorname{\mathcal{D}}& \operatorname{\mathcal{D}}_1, \ar [l] } \]

where the vertical maps are equivalences of $\infty $-categories. If $U$ and $V$ are isofibrations, then the induced map $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1} \rightarrow \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$ is an equivalence of $\infty $-categories.

Corollary 4.5.2.25. Suppose we are given a categorical pullback square of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [d]^-{U} \ar [r] & \widetilde{\operatorname{\mathcal{D}}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}, } \]

where $U$ and $V$ are isofibrations. Let $C \in \operatorname{\mathcal{C}}$ be an object having image $D = F(C)$. Then the induced map

\[ \widetilde{\operatorname{\mathcal{C}}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} = \widetilde{\operatorname{\mathcal{D}}}_{D} \]

is an equivalence of $\infty $-categories.

Proof. Apply Corollary 4.5.2.24 in the special case $\operatorname{\mathcal{C}}_1 = \{ C\} $ and $\operatorname{\mathcal{D}}_1 = \{ D\} $. $\square$

Corollary 4.5.2.26. Suppose we are given a diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [d]^-{U} \ar [r]^{ \widetilde{F}} & \widetilde{\operatorname{\mathcal{D}}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}, } \]

where $U$ and $V$ are isofibrations and the functors $F$ and $\widetilde{F}$ are equivalences of $\infty $-categories. Let $C \in \operatorname{\mathcal{C}}$ be an object having image $D = F(C)$. Then the induced map

\[ \widetilde{\operatorname{\mathcal{C}}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} = \widetilde{\operatorname{\mathcal{D}}}_{D} \]

is an equivalence of $\infty $-categories.

Warning 4.5.2.27. Suppose we are given a commutative diagram of simplicial sets

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

where $q$ and $q'$ are Kan fibrations and $f$ is a homotopy equivalence. By virtue of Proposition 3.2.8.1, the following conditions are equivalent:

$(1)$

The morphism $f'$ is a homotopy equivalence of Kan complexes.

$(2)$

For each vertex $s' \in S'$ having image $s = f(s') \in S$, the induced map of fibers $X'_{s'} \rightarrow X_{s}$ is a homotopy equivalence of Kan complexes.

Corollary 4.5.2.26 can be regarded as a generalization of the implication $(1) \Rightarrow (2)$, where we allow $\infty $-categories in place of Kan complexes and isofibrations in place of Kan fibrations. Beware that the implication $(2) \Rightarrow (1)$ does not generalize. For example, we have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^1 \ar [r] \ar [d] & \Delta ^1 \ar [d]^{ \operatorname{id}} \\ \Delta ^1 \ar [r]^-{\operatorname{id}} & \Delta ^1, } \]

where the vertical maps are isofibrations, the bottom horizontal map is an isomorphism, and the upper horizontal map restricts to an isomorphism on each fiber, but is nevertheless not an equivalence of $\infty $-categories.

Corollary 4.5.2.28. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an isofibration of $\infty $-categories, let $B \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $f: A \rightarrow B$ be a categorical equivalence of simplicial sets. Then precomposition with $f$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{C}}}(A,\operatorname{\mathcal{E}})$.

Proof. Apply Corollary 4.5.2.26 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B, \operatorname{\mathcal{E}}) \ar [r]^-{ \circ f} \ar [d]^{U \circ } & \operatorname{Fun}(A, \operatorname{\mathcal{E}}) \ar [d]^{U \circ } \\ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [r]^-{\circ f} & \operatorname{Fun}(A, \operatorname{\mathcal{C}}); } \]

note that the vertical maps are isofibrations (Corollary 4.4.5.6) and the horizontal maps are equivalences of $\infty $-categories (Proposition 4.5.3.8). $\square$

Corollary 4.5.2.29. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories, let $A \subseteq B$ be simplicial sets, and suppose we are given a diagram $A \rightarrow \operatorname{\mathcal{C}}$. Then postcomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{A/}(B, \operatorname{\mathcal{D}})$.

Proof. Apply Corollary 4.5.2.26 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \ar [r]^-{ F \circ } \ar [d] & \operatorname{Fun}(B, \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \ar [r]^-{F \circ } & \operatorname{Fun}(A, \operatorname{\mathcal{D}}); } \]

note that the vertical maps are isofibrations by virtue of Corollary 4.4.5.3 and the horizontal maps are equivalences by virtue of Remark 4.5.1.16. $\square$

Remark 4.5.2.30 (Categorical Pullback Squares of Simplicial Sets). Suppose we are given a commutative diagram of simplicial sets

4.28
\begin{equation} \begin{gathered}\label{equation:general-categorical-pullback} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_{1} \ar [r] & X. } \end{gathered} \end{equation}

Applying Proposition 4.1.3.2 repeatedly, we can enlarge 4.28 to a cubical diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [rr] \ar [dd] \ar [dr] & & X_{0} \ar [dd] \ar [dr] & \\ & \operatorname{\mathcal{C}}_{01} \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}_{0} \ar [dd] \\ X_{1} \ar [rr] \ar [dr] & & X \ar [dr] & \\ & \operatorname{\mathcal{C}}_{1} \ar [rr] & & \operatorname{\mathcal{C}}, } \]

where the diagonal maps are inner anodyne and the front face

4.29
\begin{equation} \begin{gathered}\label{equation:general-categorical-pullback2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_{1} \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

is a diagram of $\infty $-categories. Let us say that that the diagram of simplicial sets (4.28) is a categorical pullback square if the diagram of $\infty $-categories (4.29) is a categorical pullback square, in the sense of Definition 4.5.2.7. Using Proposition 4.5.2.17, it is not difficult to show that this condition depends only on the original diagram (for a more general statement, see Proposition 7.5.5.13). Beware that this more general notion of categorical pullback diagram can be badly behaved: for example, it does not satisfy the analogue of Proposition 4.5.2.20 (see Warning 4.5.5.12).