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 4.4.1.14). 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.
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 1.4.6.10): 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.
Example 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 ordinary categories. Then the homotopy fiber product $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0) \times ^{\mathrm{h}}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_1)$ can be identified with the nerve of a category $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$, which can be described concretely as follows:
The objects of $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ are 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 $\operatorname{\mathcal{C}}$.
A morphism from $(C_0, C_1, e)$ to $(C'_0, C'_1, e')$ is a pair $(f_0, f_1)$, where $f_0: C_0 \rightarrow C'_0$ is a morphism in the category $\operatorname{\mathcal{C}}_0$, $f_1: C_1 \rightarrow C'_1$ is a morphism in the category $\operatorname{\mathcal{C}}_1$, and the diagram
\[ \xymatrix { C_0 \ar [r]^{f_0} \ar [d]^{e}_{\sim } & C'_0 \ar [d]^{e'}_{\sim } \\ C_1 \ar [r]^{ f_1 } & C'_1 } \]
commutes in the category $\operatorname{\mathcal{C}}$.
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 $\operatorname{\mathcal{C}}$.
Definition 4.5.2.8. 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.
Proposition 4.5.2.10. 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.
Proof.
Combine Corollary 3.4.1.6 with Examples 4.5.2.3 and Example 4.5.1.13.
$\square$
Variant 4.5.2.11. Suppose we are given a commutative diagram of $\infty $-categories
4.23
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square55} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^{q} \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}
where $\operatorname{\mathcal{C}}$ is a Kan complex. If (4.23) is a categorical pullback square, then it is also a homotopy pullback square.
Proof.
By assumption, the induced map $\operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1}$ is an equivalence of $\infty $-categories, and therefore a weak homotopy equivalence of simplicial sets (Remark 4.5.3.4). The desired result now follows from the criterion of Corollary 3.4.1.6.
$\square$
In more general situations, the notions of homotopy pullback square and categorical pullback square are distinct:
Exercise 4.5.2.12. 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 4.5.2.13. 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 4.5.2.14. A commutative diagram of $\infty $-categories
4.24
\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
4.25
\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 (4.24) 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 4.5.2.6 and 4.5.2.7, 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 (4.25). The desired result now follows from the criterion of Corollary 3.4.1.6.
$\square$
We now apply Proposition 4.5.2.14 to deduce some formal properties of the notion of categorical pullback square.
Proposition 4.5.2.16. 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.
Proof.
Combine Proposition 4.5.2.14 with Remark 3.4.1.7.
$\square$
Proposition 4.5.2.17 (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.
Proof.
Combine Propositions 4.5.2.14 and 3.4.1.9.
$\square$
Proposition 4.5.2.18 (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 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.
Proof.
Combine Propositions 4.5.2.14 and 3.4.1.11.
$\square$
Proposition 4.5.2.19 (Homotopy Invariance). Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [dr]^{ F_{01} } \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}_{0} \ar [dd] \ar [dr]^{F_0} & \\ & \operatorname{\mathcal{D}}_{01} \ar [rr] \ar [dd] & & \operatorname{\mathcal{D}}_{0} \ar [dd] \\ \operatorname{\mathcal{C}}_{1} \ar [rr] \ar [dr]^{ F_1 } & & \operatorname{\mathcal{C}}\ar [dr]^{F} & \\ & \operatorname{\mathcal{D}}_{1} \ar [rr] & & \operatorname{\mathcal{D}}, } \]
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 4.5.2.14 with Corollary 3.4.1.12.
$\square$
Corollary 4.5.2.20. 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 4.5.2.21. Suppose we are given a commutative diagram of $\infty $-categories
4.26
\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 (4.26) is a categorical pullback square if and only if $F'$ is an equivalence of $\infty $-categories.
Proof.
Combine Proposition 4.5.1.22, Proposition 4.5.2.14, and Corollary 3.4.1.5.
$\square$
Corollary 4.5.2.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let
\[ \delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\{ 0\} , \operatorname{\mathcal{D}})} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}}) \]
be map induced by the diagonal embedding $c: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$. Then $\delta $ is fully faithful, and its essential image is the homotopy fiber product $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ of Construction 4.5.2.1.
Proof.
Let us identify the objects of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ with triples $(C,D,u)$, where $C$ is an object of $\operatorname{\mathcal{C}}$, $D$ is an object of $\operatorname{\mathcal{D}}$, and $u: F(C) \rightarrow D$ is a morphism in $\operatorname{\mathcal{D}}$. By definition, $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is the full subcategory of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ spanned by those triples $(C,D,u)$ where $u$ is an isomorphism in $\operatorname{\mathcal{D}}$. The functor $\delta $ is given on objects by the formula $\delta (C) = ( C, F(C), \operatorname{id}_{ F(C) } )$, and therefore factors through $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. To complete the proof, it will suffice to show that the functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories. Equivalently, we wish to show that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^{\operatorname{id}} \ar [d]^{F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\ar [r]^{\operatorname{id}} & \operatorname{\mathcal{D}}} \]
is a categorical pullback square, which is a special case of Proposition 4.5.2.21.
$\square$
Corollary 4.5.2.23. Let $f: K \rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Then $f$ factors as a composition $K \xrightarrow {j} \operatorname{\mathcal{C}}\xrightarrow {U} \operatorname{\mathcal{D}}$, where $U$ is an isofibration of $\infty $-categories and $j$ is both a monomorphism and a categorical equivalence.
Proof.
Using Proposition 4.1.3.2, we can factor $f$ as a composition $K \xrightarrow {i} \operatorname{\mathcal{K}}\xrightarrow {F} \operatorname{\mathcal{D}}$, where $i$ is inner anodyne and $F$ is an inner fibration. Note that the simplicial set $\operatorname{\mathcal{K}}$ is an $\infty $-category (Remark 4.1.1.9), and that $i$ is a categorical equivalence of simplicial sets (Corollary 4.5.3.14). We may therefore replace $f$ by $F$, and thereby reduce to the special case where $K = \operatorname{\mathcal{K}}$ is an $\infty $-category. Let $\operatorname{\mathcal{C}}$ denote the homotopy fiber product $\operatorname{\mathcal{K}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Then $F$ factors as a composition
\[ \operatorname{\mathcal{K}}\xrightarrow { \delta } \operatorname{\mathcal{K}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\xrightarrow {U} \operatorname{\mathcal{D}}, \]
where the diagonal embedding $\delta $ is an equivalence of $\infty $-categories (Corollary 4.5.2.22) and $U$ is an isofibration (see Remark 4.5.2.2).
$\square$
Proposition 4.5.2.26. Suppose we are given a commutative diagram of $\infty $-categories
4.27
\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 (4.27) 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 4.5.2.14 with Example 3.4.1.3, we see that (4.27) 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 4.4.3.19, 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 4.5.2.27. Suppose we are given a pullback diagram of $\infty $-categories
4.28
\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 (4.28) is a categorical pullback square.
Corollary 4.5.2.28. 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 4.5.2.27.
$\square$
Corollary 4.5.2.29. 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.
Proof.
Combine Corollary 4.5.2.27 with Proposition 4.5.2.21.
$\square$
Corollary 4.5.2.30. 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.
Proof.
Combine Corollaries 4.5.2.20 and 4.5.2.28.
$\square$
Corollary 4.5.2.31. 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 4.5.2.30 in the special case $\operatorname{\mathcal{C}}_1 = \{ C\} $ and $\operatorname{\mathcal{D}}_1 = \{ D\} $.
$\square$
Corollary 4.5.2.32. 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.
Proof.
Combine Proposition 4.5.2.21 with Corollary 4.5.2.31.
$\square$
Warning 4.5.2.33. Suppose we are given a commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r]^-{f'} \ar [d]^{q'} & X \ar [d]^{q} \\ S' \ar [r]^-{f} & S, } \]
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 4.5.2.32 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 4.5.2.34. 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 4.5.2.32 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 4.5.2.35. 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 4.5.2.32 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.
$\square$
