7.6.4 Examples of Pullback and Pushout Squares
We now give some examples of $\infty $-categorical pullback diagrams.
Proposition 7.6.4.1. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and let $\sigma :$
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \]
be a commutative diagram in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The composite map
\[ \Delta ^1 \times \Delta ^1 \xrightarrow { \operatorname{N}_{\bullet }(\sigma ) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \]
is a pullback square in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (in the sense of Definition 7.6.3.1).
- $(2)$
For every object $Y \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X_{01} )_{\bullet } \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, X_0 )_{\bullet } \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, X_1 )_{\bullet } \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } } \]
is a homotopy pullback square (in the sense of Definition 3.4.1.1).
Proof.
Combine Corollary 7.5.4.6 with Proposition 7.5.4.13.
$\square$
Example 7.6.4.2. A (strictly) commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_{1} \ar [r] & X } \]
is a homotopy pullback square (in the sense of Definition 3.4.1.1) if and only if the induced diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}$ is a pullback square in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$. This follows by combining Propositions 7.5.4.13 and 7.5.4.5.
Example 7.6.4.3. A (strictly) commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_{1} \ar [r] & A_{01} } \]
is a homotopy pushout square (in the sense of Definition 3.4.2.1) if and only if the induced diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}$ is a pushout square in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$. This follows by combining Corollaries 7.5.7.7 and 7.5.7.9.
Example 7.6.4.4. A (strictly) 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 (in the sense of Definition 4.5.2.8) if and only if the induced diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{QCat}) = \operatorname{\mathcal{QC}}$ is a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$. This follows by combining Corollaries 7.5.5.8 and 7.5.5.10.
Example 7.6.4.5. A (strictly) commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_{1} \ar [r] & \operatorname{\mathcal{C}}_{01} } \]
is a categorical pushout square (in the sense of Definition 4.5.4) if and only if the induced diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{QCat}) = \operatorname{\mathcal{QC}}$ is a pushout square in the $\infty $-category $\operatorname{\mathcal{QC}}$. This follows by combining Corollaries 7.5.8.5 and 7.5.8.9.
Recall that the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ admits small limits and colimits (Corollary 7.4.5.6). In particular, if $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ are morphisms of Kan complexes, then there exists a pullback diagram $\sigma :$
\[ \xymatrix@C =50pt@R=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X } \]
in the $\infty $-category $\operatorname{\mathcal{S}}$. However, it is not always possible to obtain $\sigma $ from a commutative diagram in the ordinary category $\operatorname{Kan}$. It will therefore be useful to have a generalization of Proposition 7.6.4.1, which applies to homotopy coherent squares.
We can summarize this data in a diagram
\[ \xymatrix@C =100pt@R=100pt{ X_{01} \ar [r]^-{g_0} \ar [d]_-{g_1} \ar [dr]^-{h} & X_0 \ar [d]^-{f_0} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>_-{\alpha _0} \\ X_1 \ar [r]_-{ f_1 } \ar@ {=>}[]+<20pt,20pt>;+<45pt,45pt>^-{\alpha _1} & X. } \]
Here we can regard $(a)$ and $(b)$ as supplying a (potentially) non-commutative square diagram in the category $\operatorname{\mathcal{C}}$, and $(c)$ as supplying a witness to the fact that it commutes up to homotopy.
Example 7.6.4.7 (Square Diagrams in $\operatorname{\mathcal{QC}}$). 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. Using Remark 7.6.4.6, we see that the data of a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^-{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{F_1} & \operatorname{\mathcal{C}}} \]
in the $\infty $-category $\operatorname{\mathcal{QC}}$ is equivalent to the data of an $\infty $-category $\operatorname{\mathcal{C}}_{01}$ equipped with functors
\[ G_0: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_0 \quad \quad G_1: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_1 \quad \quad H: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}} \]
together with natural isomorphisms $\alpha _0: (F_0 \circ G_0) \xrightarrow {\sim } H$ and $\alpha _1: (F_1 \circ G_1) \xrightarrow {\sim } H$. In this case, we can identify the data of the tuple $(G_0, \alpha _0, G_1, \alpha _1, H)$ with a single functor of $\infty $-categories
\[ G: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{C}}_1 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}). \]
Proposition 7.6.4.8. Suppose we are given a commutative diagram
7.62
\begin{equation} \begin{gathered}\label{equation:all-squares-in-QCat} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{0} \ar [d]^-{ F_0 } \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{ F_1 } & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{QC}}$, corresponding to a functor
\[ G: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{C}}_1 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}). \]
Then (7.62) is a pullback square in $\operatorname{\mathcal{QC}}$ if and only if the functor $G$ is an equivalence of $\infty $-categories.
Proof.
Let us identify the diagram (7.62) with a functor of simplicial categories $\mathscr {F}: \operatorname{Path}[ [1] \times [1] ]_{\bullet } \rightarrow \operatorname{QCat}$. Using Corollary 4.5.2.23, we can factor the functor $F_0$ as a composition $\operatorname{\mathcal{C}}_0 \xrightarrow { T } \operatorname{\mathcal{C}}'_{0} \xrightarrow { F'_{0} } \operatorname{\mathcal{C}}$, where $T$ is an equivalence of $\infty $-categories and $F'_{0}$ is an isofibration. Let $\operatorname{\mathcal{C}}'_{01}$ denote the iterated homotopy fiber product $\operatorname{\mathcal{C}}'_{0} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{C}}_1 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})$. Then Example 7.6.4.7 supplies a commutative diagram
7.63
\begin{equation} \begin{gathered}\label{equation:all-squares-in-QCat2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}'_{0} \ar [d]^-{F'_{0}} \\ \operatorname{\mathcal{C}}_{1} \ar [r]^-{F_1} & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{QC}}$, which we view as a functor of simplicial categories $\mathscr {F}': \operatorname{Path}[ [1] \times [1] ]_{\bullet } \rightarrow \operatorname{QCat}$. The morphisms $G$ and $T$ determine a natural transformation of simplicial functors $\mathscr {F} \rightarrow \mathscr {F}'$, which induces a natural transformation from the diagram (7.62) to the diagram (7.63) in the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{QC}})$. By virtue of Corollary 4.5.2.20, this natural transformation is an isomorphism of diagrams if and only if the functor $G$ is an equivalence of $\infty $-categories. Consequently, Proposition 7.6.4.8 is equivalent to the assertion that (7.63) is a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$ (see Proposition 7.1.2.13).
Note that we have a (strictly) commutative diagram of simplicial sets
7.64
\begin{equation} \begin{gathered}\label{equation:all-squares-in-QCat3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'_{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1} \ar [r] \ar [d] & \operatorname{\mathcal{C}}'_{0} \ar [d]^-{F'_{0}} \\ \operatorname{\mathcal{C}}_{1} \ar [r]^-{F'_1} & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}
which determines a subfunctor $\mathscr {F}'' \subseteq \mathscr {F}'$. Since $F'_0$ is an isofibration, it follows from Corollaries 4.5.2.28, 4.5.2.22, and 4.5.2.29 that the inclusion maps
\begin{eqnarray*} \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 \\ & \hookrightarrow & \operatorname{\mathcal{C}}'_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} ( \operatorname{\mathcal{C}}_1 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \end{eqnarray*}
are equivalences of $\infty $-categories. Consequently, the inclusion $\mathscr {F}'' \hookrightarrow \mathscr {F}'$ is a levelwise categorical equivalence of simplicial functors and therefore induces an isomorphism from the diagram ( 7.64 ) to the diagram ( 7.63 ) in the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}})$. By virtue of Proposition 7.1.2.13, it will suffice to show that the diagram (7.64) is a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$. This is a special case of Example 7.6.4.4, since (7.64) is a categorical pullback square (see Corollary 4.5.2.27).
$\square$
Corollary 7.6.4.9. 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, let let $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ denote the homotopy fiber product of Construction 4.5.2.1, and let
\[ G_0: \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_0 \quad \quad G_1: \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_1 \]
denote the projection maps, so that we have a canonical isomorphism $\alpha : F_0 \circ G_0 \rightarrow F_1 \circ G_1$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1, \operatorname{\mathcal{C}})$. Then the diagram
\[ \xymatrix@C =100pt@R=100pt{ \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1 \ar [r]^-{G_0} \ar [d]_-{G_1} \ar [dr]^-{F_1 \circ G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^-{F_0} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>_-{\alpha } \\ \operatorname{\mathcal{C}}_1 \ar [r]_-{ F_1 } \ar@ {=>}[]+<20pt,20pt>;+<45pt,45pt>^-{\operatorname{id}} & \operatorname{\mathcal{C}}} \]
corresponds to a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$. In particular, $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a fiber product of $\operatorname{\mathcal{C}}_0$ with $\operatorname{\mathcal{C}}_1$ over $\operatorname{\mathcal{C}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$.
Proof.
By virtue of Proposition 7.6.4.8, it will suffice to show that the inclusion
\[ \delta : \operatorname{\mathcal{C}}_{1} \simeq \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}_{1} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}} \]
induces an equivalence of homotopy fiber products
\[ \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1} \hookrightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} ( \operatorname{\mathcal{C}}_1 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}). \]
This is a special case of Corollary 4.5.2.20, since $\delta $ is an equivalence of $\infty $-categories (Corollary 4.5.2.22).
$\square$
Corollary 7.6.4.10. Suppose we are given a commutative diagram
7.65
\begin{equation} \begin{gathered}\label{equation:all-squares-in-SSet} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_{0} \ar [d] \\ X_1 \ar [r] & X } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{S}}$, classified by a map of Kan complexes
\[ g: X_{01} \rightarrow X_{0} \times ^{\mathrm{h}}_{ X} ( X_1 \times ^{\mathrm{h}}_{X} X ). \]
Then (7.65) is a pullback square in $\operatorname{\mathcal{S}}$ if and only if $g$ is a homotopy equivalence.
Proof.
Combine Propositions 7.6.4.8 and 7.4.5.1.
$\square$
Corollary 7.6.4.11. Let $n$ be an integer and suppose we are given a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^{f'} & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]
in the $\infty $-category $\operatorname{\mathcal{S}}$. If $f$ is $n$-truncated, then $f'$ is $n$-truncated. If $f$ is $n$-connective, then $f'$ is $n$-connective.
Proof.
Combine Corollaries 7.6.4.10, 3.5.9.12, and 3.5.1.25.
$\square$
Example 7.6.4.12. Let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be morphisms of Kan complexes. Applying the construction of Corollary 7.6.4.9, we obtain a pullback square
\[ \xymatrix@R =50pt@C=50pt{ X_0 \times ^{\mathrm{h}}_{X} X_1 \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X } \]
in the $\infty $-category $\operatorname{\mathcal{S}}$.
Exercise 7.6.4.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, so that Corollary 7.6.4.9 supplied an identification of $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{C}}$ with the fiber product of $\operatorname{\mathcal{C}}$ with itself over $\operatorname{\mathcal{D}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$. Show that, under this identification, the relative diagonal of $F$ (in the sense of Notation 7.6.3.18) is represented by the inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{C}}$. Moreover, if $F$ is an isofibration, then we can replace the homotopy fiber product $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ with the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ (formed in the ordinary category of simplicial sets); see Corollary 4.5.2.27.
Corollary 7.6.4.14. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, and suppose we are given a commutative diagram $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ \xymatrix@C =100pt@R=100pt{ X_{01} \ar [r]^-{g_0} \ar [d]_-{g_1} \ar [dr]^-{h} & X_0 \ar [d]^-{f_0} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>_-{\alpha _0} \\ X_1 \ar [r]_-{ f_1 } \ar@ {=>}[]+<20pt,20pt>;+<45pt,45pt>^-{\alpha _1} & X } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Remark 7.6.4.6). Then $\sigma $ is a pullback square in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Y \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X_{01})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X_0)_{\bullet } \times ^{\mathrm{h}}_{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } } ( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X_1)_{\bullet } \times ^{\mathrm{h}}_{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } }\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } ) \]
is a homotopy equivalence of Kan complexes.
Proof.
Combine Corollary 7.6.4.10 with Proposition 7.4.5.16.
$\square$