3.4.1 Homotopy Pullback Squares
We begin by formulating the notion of a homotopy pullback square for general simplicial sets.
Definition 3.4.1.1. A commutative diagram of simplicial sets
3.36
\begin{equation} \label{diagram:homotopy-pullback-square0} \begin{gathered} \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 homotopy pullback square if, for every factorization $q = q' \circ w$ where $w: X_0 \rightarrow X'_0$ is a weak homotopy equivalence and $q': X'_0 \rightarrow X$ is a Kan fibration, the induced map $X_{01} \rightarrow X'_0 \times _{X} X_1$ is a weak homotopy equivalence.
To verify the condition of Definition 3.4.1.1 in general, it suffices to consider a single factorization $q = q' \circ w$:
Proposition 3.4.1.2. Suppose we are given a commutative diagram of simplicial sets
3.37
\begin{equation} \label{diagram:homotopy-pullback-square1} \begin{gathered} \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}
Suppose that $q$ factors as a composition $X_0 \xrightarrow {w'} X'_0 \xrightarrow {q'} X$, where $w'$ is a weak homotopy equivalence and $q'$ is a Kan fibration. Then (3.37) is a homotopy pullback square if and only if the induced map $\rho ': X_{01} \rightarrow X'_{0} \times _{X} X_1$ is a weak homotopy equivalence.
Proof.
Suppose that $q$ admits another factorization $X_0 \xrightarrow {w''} X''_0 \xrightarrow { q''} X$, where $w''$ is a weak homotopy equivalence and $q''$ is a Kan fibration. We wish to show that $\rho '$ is a weak homotopy equivalence if and only if the induced map $\rho '': X_{01} \rightarrow X''_{0} \times _{X} X_1$ is a weak homotopy equivalence. To prove this equivalence, we may assume without loss of generality that $w'$ is anodyne (since this can always be arranged using Proposition 3.1.7.1). In this case, the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [d]^{ w' } \ar [r]^-{w''} & X''_0 \ar [d]^{q''} \\ X'_0 \ar@ {-->}[ur]^{u} \ar [r]^-{q'} & X } \]
admits a solution $u: X'_0 \rightarrow X''_0$ (Remark 3.1.2.7). Since $w'$ and $w''$ are weak homotopy equivalences, the equality $w'' = u \circ w'$ guarantees that $u$ is also a weak homotopy equivalence (Remark 3.1.6.16), so that the map $X'_0 \times _{X} X_1 \rightarrow X''_0 \times _{X} X_1$ is a weak homotopy equivalence by virtue of Proposition 3.4.0.2.
$\square$
Example 3.4.1.3. Suppose we are given a commutative diagram of simplicial sets
3.38
\begin{equation} \label{diagram:homotopy-pullback-square3} \begin{gathered} \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}
where $q$ is a Kan fibration. Applying Proposition 3.4.1.2 to the factorization $q = q \circ \operatorname{id}_{X_0}$, we see that (3.38) is a homotopy pullback square if and only if the induced map $X_{01} \rightarrow X_0 \times _{X} X_{1}$ is a weak homotopy equivalence. In particular, if (3.38) is a pullback diagram, then it is also a homotopy pullback diagram. Beware that this conclusion is generally false when $q$ is not a Kan fibration.
Example 3.4.1.4. Suppose we are given a commutative diagram of simplicial sets
3.39
\begin{equation} \label{diagram:homotopy-pullback-square2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^{q'} & X \ar [d]^{q} \\ S' \ar [r] & S, } \end{gathered}\end{equation}
where $q$ and $q'$ are Kan fibrations. Then (3.39) is a homotopy pullback square if and only if, for each vertex $s' \in S'$ having image $s \in S$, the induced map of fibers $X'_{s'} \rightarrow X_{s}$ is a homotopy equivalence of Kan complexes. This is essentially a restatement of Proposition 3.3.7.1 (by virtue of Proposition 3.4.1.2).
Corollary 3.4.1.5. Suppose we are given a commutative diagram of simplicial sets
3.40
\begin{equation} \label{diagram:homotopy-pullback-square5} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d]^{q'} & X_0 \ar [d]^{q} \\ X_1 \ar [r] & X, } \end{gathered}\end{equation}
where $q$ is a weak homotopy equivalence. Then (3.40) is a homotopy pullback square if and only if $q'$ is a weak homotopy equivalence.
Proof.
Apply Proposition 3.4.1.2 to the factorization $q = \operatorname{id}_{X} \circ q$.
$\square$
Corollary 3.4.1.6. Suppose we are given a commutative diagram of simplicial sets
3.41
\begin{equation} \label{diagram:homotopy-pullback-square55} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d]^{q'} & X_0 \ar [d]^{q} \\ X_1 \ar [r] & X, } \end{gathered}\end{equation}
where $X$ is a Kan complex. Then (3.41) is a homotopy pullback square if and only if the induced map
\[ \theta : X_{01} \rightarrow X_0 \times _{X} X_1 \hookrightarrow X_{0} \times _{X}^{\mathrm{h}} X_{1} \]
is a weak homotopy equivalence.
Proof.
Using Proposition 3.1.7.1, we can factor $q$ as a composition $X_0 \xrightarrow {w} X'_0 \xrightarrow {q'} X$, where $w$ is a weak homotopy equivalence and $q'$ is Kan fibration. We then have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r]^-{\theta } \ar [d]^{\rho } & X_{0} \times _{X}^{\mathrm{h}} X_{1} \ar [d] \\ X'_{0} \times _{X} X_{1} \ar [r] & X'_{0} \times _{X}^{\mathrm{h}} X_{1}, } \]
where the bottom horizontal map is a weak homotopy equivalence (Proposition 3.4.0.7) and the right vertical map is also a weak homotopy equivalence (Proposition 3.4.0.9). It follows that $\theta $ is a weak homotopy equivalence if and only if $\rho $ is a weak homotopy equivalence. By virtue of Proposition 3.4.1.2, this is equivalent to the requirement that the diagram (3.41) is a homotopy pullback square.
$\square$
Warning 3.4.1.8. For a general pair of morphisms $f_0: X_0 \rightarrow X$, $f_1: X_1 \rightarrow X$ in the category of simplicial sets, there need not exist a homotopy pullback square
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]
For example, if $f_0: \{ 0\} \hookrightarrow \Delta ^1$ and $f_1: \{ 1\} \hookrightarrow \Delta ^1$ are the inclusion maps, then the existence of a commutative diagram
3.42
\begin{equation} \label{diagram:homotopy-pullback-square4} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & \{ 0\} \ar [d]^{f_0} \\ \{ 1\} \ar [r]^-{f_1} & \Delta ^1 } \end{gathered}\end{equation}
guarantees that the simplicial set $X_{01}$ is empty, in which case (3.42) is not a homotopy pullback square.
Note that Definition 3.4.1.1 is a priori asymmetric: it involves replacing the map $f_0: X_0 \rightarrow X$ by a Kan fibration, but leaving the map $f_1: X_1 \rightarrow X$ unchanged. However, this turns out to be irrelevant.
Proposition 3.4.1.9 (Symmetry). A commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{f_0} \\ X_1 \ar [r]^-{f_1} & X } \]
is a homotopy pullback square if and only if the transposed diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_1 \ar [d]^{f_1} \\ X_0 \ar [r]^-{f_0} & X } \]
is a homotopy pullback square.
Proof.
Using Proposition 3.1.7.1, we can choose factorizations
\[ X_0 \xrightarrow {w_0} X'_0 \xrightarrow {f'_0} S \quad \quad X_1 \xrightarrow {w_1} X'_1 \xrightarrow {f'_1} S \]
of $f_0$ and $f_1$, where both $f'_0$ and $f'_1$ are Kan fibrations and both $w_0$ and $w_1$ are weak homotopy equivalences. We have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r]^-{u} \ar [d]^{v} & X_0 \times _{X} X'_1 \ar [r] \ar [d]^{v'} & X_0 \ar [d]^{w_0} \\ X'_0 \times _{X} X_1 \ar [r]^-{u'} \ar [d] & X'_0 \times _{X} X'_1 \ar [r] \ar [d] & X'_0 \ar [d]^{f'_0} \\ X_1 \ar [r]^-{w_1} & X'_1 \ar [r]^-{ f'_1} & X. } \]
We wish to show that $u$ is a weak homotopy equivalence if and only if $v$ is a weak homotopy equivalence (see Proposition 3.4.1.2). This follows from the two-out-of-three property (Remark 3.1.6.16), since the morphisms $u'$ and $v'$ are weak homotopy equivalences by virtue of Corollary 3.3.7.4.
$\square$
Proposition 3.4.1.11 (Transitivity). Suppose we are given a commutative diagram of simplicial sets
3.43
\begin{equation} \begin{gathered}\label{equation:composite-homotopy-pullback} \xymatrix@R =50pt@C=50pt{ Z \ar [r] \ar [d]^{h} & Y \ar [r] \ar [d]^{g} & X \ar [d]^{f} \\ U \ar [r] & T \ar [r] & S } \end{gathered} \end{equation}
where the right half of (3.43) is a homotopy pullback square. Then the left half of (3.43) is a homotopy pullback square if and only if the outer rectangle is a homotopy pullback square.
Proof.
By virtue of Proposition 3.1.7.1, the morphism $f$ factors as a composition $X \xrightarrow {w_ X} X' \xrightarrow {f'} S$, where $f'$ is a Kan fibration and $w_{X}$ is a weak homotopy equivalence. Set $Y' = T \times _{S} X'$, so that $g$ factors as a composition $Y \xrightarrow { w_{Y} } Y' \xrightarrow {g'} T$ where $g'$ is a Kan fibration. Since the right half of (3.43) is a homotopy pullback square, the morphism $w_{Y}$ is a weak homotopy equivalence. Applying Proposition 3.4.1.2, we see that both conditions are equivalent to the requirement that the induced map $Z \rightarrow U \times _{T} Y' \simeq U \times _{S} X'$ is a weak homotopy equivalence.
$\square$
Corollary 3.4.1.12 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@C =50pt{ X_{01} \ar [rr] \ar [dd] \ar [dr]^-{w_{01}} & & X_{0} \ar [dd] \ar [dr]^-{ w_{0} } & \\ & Y_{01} \ar [rr] \ar [dd] & & Y_0 \ar [dd] \\ X_1 \ar [rr] \ar [dr]^-{ w_1} & & X \ar [dr]^-{w} & \\ & Y_1 \ar [rr] & & Y, } \]
where the morphisms $w_0$, $w_1$, and $w$ are weak homotopy equivalences. Then any two of the following conditions imply the third:
- $(1)$
The back face
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \]
is a homotopy pullback square.
- $(2)$
The front face
\[ \xymatrix@R =50pt@C=50pt{ Y_{01} \ar [r] \ar [d] & Y_0 \ar [d] \\ Y_1 \ar [r] & Y } \]
is a homotopy pullback square.
- $(3)$
The morphism $w_{01}: X_{01} \rightarrow Y_{01}$ is a weak homotopy equivalence of simplicial sets.
Proof.
Using Corollary 3.4.1.5, we see that the bottom square in the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] \ar [d]^{w_1} & X \ar [d]^{w} \\ Y_1 \ar [r] & Y, } \]
is a homotopy pullback square. Applying Propositions 3.4.1.11 and 3.4.1.9, we see that $(1)$ is equivalent to the following:
- $(1')$
The diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ Y_1 \ar [r] & Y } \]
is a homotopy pullback square.
If condition $(3)$ is satisfied, then the equivalence $(1') \Leftrightarrow (2)$ is a special case of Remark 3.4.1.10. Conversely, if $(1')$ and $(2)$ are satisfied, then Propositions 3.4.1.11 and 3.4.1.9 guarantee that the upper half of the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d]^{w_{01}} & X_0 \ar [d]^{w_0} \\ Y_{01} \ar [r] \ar [d] & Y_0 \ar [d] \\ Y_1 \ar [r] & Y } \]
is a homotopy pullback square, so that $w_{01}$ is a weak homotopy equivalence by virtue of Corollary 3.4.1.5.
$\square$
Suppose we are given a commutative diagram of Kan complexes $\sigma :$
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r]^-{f_1} & X. } \]
It follows from Corollary 3.4.1.12 that the condition that $\sigma $ is a homotopy pullback square depends only on the homotopy type of $\sigma $ as an object of the diagram category $\operatorname{Fun}( [1] \times [1], \operatorname{Kan})$. Beware that it does not depend only on the image of $\sigma $ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.
Example 3.4.1.13. Let $X$ be a Kan complex containing a vertex $x \in X$, let $\Omega X$ denote the loop space $\{ x\} \times ^{\mathrm{h}}_{X} \{ x\} $, and let $P$ denote the path space $X \times _{X}^{\mathrm{h}} \{ x\} $, and let $\iota : \Omega X \hookrightarrow P$ be the inclusion map. We then have a pullback diagram of Kan complexes
3.44
\begin{equation} \begin{gathered}\label{equation:path-space-counterexample} \xymatrix@R =50pt@C=50pt{ \Omega X \ar [r]^-{\iota } \ar [d] & P \ar [d]^{\operatorname{ev}_0} \\ \{ x\} \ar [r] & X, } \end{gathered} \end{equation}
where $\operatorname{ev}_0$ is given by evaluation at the vertex $0 \in \Delta ^1$. Since $\operatorname{ev}_0$ is a Kan fibration, the diagram (3.44) is also a homotopy pullback square (Example 3.4.1.3). Note that the Kan complex $P$ is contractible, so that $\iota $ is homotopic to the constant map $\iota ': \Omega X \rightarrow P$ carrying $\Omega X$ to the constant path $\operatorname{id}_{x}$. However, the commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \Omega X \ar [r]^-{\iota '} \ar [d] & P \ar [d]^{\operatorname{ev}_0} \\ \{ x\} \ar [r] & X } \]
is never a homotopy pullback square unless the Kan complex $\Omega X$ is contractible (again by Example 3.4.1.3).
Proposition 3.4.1.14 (Summands). Suppose we are given a homotopy pullback square of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r]^-{u} \ar [d]^{v} & X_0 \ar [d]^{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]
Let $X'_0 \subseteq X_0$, $X'_1 \subseteq X_1$, and $X' \subseteq X$ be summands satisfying $f_0(X'_0) \subseteq X' \supseteq f_1(X'_1)$, and set $X'_{01} = u^{-1}(X'_0) \cap v^{-1}(X'_1) \subseteq X_{01}$. Then the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X'_{01} \ar [r] \ar [d] & X'_0 \ar [d] \\ X'_1 \ar [r] & X' } \]
is also a homotopy pullback square.
Proof.
Consider the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ v^{-1}( X'_1) \ar [r] \ar [d] & X_{01} \ar [r]^-{u} \ar [d]^{v} & X_0 \ar [d]^{f_0} \\ X'_1 \ar [r] & X_1 \ar [r] & X. } \]
The square on the left is a pullback diagram whose horizontal maps are Kan fibrations (Example 3.1.1.4), and is therefore a homotopy pullback square (Example 3.4.1.3). The square on the right is a homotopy pullback by assumption. Applying Proposition 3.4.1.11, we deduce that bottom half of the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X'_{01} \ar [r] \ar [d] & X'_0 \ar [d] \\ v^{-1}(X'_1) \ar [r] \ar [d] & X_0 \ar [d] \\ X'_1 \ar [r] & X } \]
is a homotopy pullback square. The top half is a pullback diagram whose vertical maps are Kan fibrations (Example 3.1.1.4), and is therefore also a homotopy pullback square (Example 3.4.1.3). Applying Proposition 3.4.1.11 again, we conclude that the outer rectangle in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X'_{01} \ar [r] \ar [d] & X'_0 \ar@ {=}[r] \ar [d] & X'_0 \ar [d] \\ X'_1 \ar [r] & X' \ar [r] & X } \]
is a homotopy pullback square. Here the square on the right is a pullback diagram of Kan fibrations (Example 3.1.1.4), and therefore a homotopy pullback. Applying Proposition 3.4.1.11 again, we conclude that the left square is a homotopy pullback, as desired.
$\square$