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3.4.1 Homotopy Pullback Squares

We begin by reformulating Construction 3.4.0.5.

Definition 3.4.1.1. A commutative diagram of simplicial sets

3.38
\begin{equation} \label{diagram:homotopy-pullback-square0} \begin{gathered} \xymatrix { Y \ar [r] \ar [d] & X \ar [d]^{f} \\ T \ar [r] & S } \end{gathered}\end{equation}

is a homotopy pullback square if there exists a factorization $f = f' \circ w$ where $f': X' \rightarrow S$ is a Kan fibration, $w: X \rightarrow X'$ is a weak homotopy equivalence, and the induced map $Y \rightarrow T \times _{S} X'$ is a weak homotopy equivalence.

Stated more informally, the diagram (3.38) is homotopy pullback square if it exhibits $Y$ as (weakly homotopy equivalent to) the homotopy fiber product $T \times _{S}^{h} X$ of Construction 3.4.0.5. This condition is actually independent of the choices involved in the construction of the homotopy fiber product, by virtue of the following:

Proposition 3.4.1.2. Suppose we are given a commutative diagram of simplicial sets

3.39
\begin{equation} \label{diagram:homotopy-pullback-square1} \begin{gathered} \xymatrix { Y \ar [r] \ar [d] & X \ar [d]^{f} \\ T \ar [r] & S. } \end{gathered}\end{equation}

The following conditions are equivalent:

$(1)$

Diagram (3.39) is a homotopy pullback square, in the sense of Definition 3.4.1.1. That is, there exists a factorization $X \xrightarrow {w} X' \xrightarrow {f'} X$ of the morphism $f$ where $f'$ is a Kan fibration, $w$ is a weak homotopy equivalence, and the induced map $Y \rightarrow T \times _{S} X'$ is a weak homotopy equivalence.

$(2)$

For every factorization $X \xrightarrow {w} X' \xrightarrow {f'} S$ of the morphism $f$, if $f'$ is a Kan fibration and $w$ is a weak homotopy equivalence, then the induced map $Y \rightarrow T \times _{S} X'$ is also a weak homotopy equivalence.

Proof. The implication $(2) \Rightarrow (1)$ follows from Proposition 3.1.6.1. To prove the converse, suppose that the morphism $f$ admits two different factorizations

\[ X \xrightarrow {w'} X' \xrightarrow {f'} S \quad \quad X \xrightarrow {w''} X'' \xrightarrow { f''} S, \]

where both $w'$ and $w''$ are weak homotopy equivalences and both $f'$ and $f''$ are Kan fibrations. We wish to show that the induced map $\rho ': Y \rightarrow T \times _{S} X'$ is a weak homotopy equivalence if and only if the induced map $\rho '': Y \rightarrow T \times _{S} X''$ is a weak homotopy equivalence. To establish this equivalence, we may assume without loss of generality that $w'$ is anodyne (since this can always be arranged using Proposition 3.1.6.1). In this case, the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{ w' } \ar [r]^{w''} & X'' \ar [d]^{f''} \\ X' \ar@ {-->}[ur]^{u} \ar [r]^{f'} & S } \]

admits a solution $u: X' \rightarrow X''$ (Remark 3.1.2.6). Since $w'$ and $w''$ are weak homotopy equivalences, the equality $w'' = u \circ w'$ guarantees that $u$ is also a weak homotopy equivalence (Remark 3.1.5.16), so that the map $T \times _{S} X' \rightarrow T \times _{S} X''$ is a weak homotopy equivalence by virtue of Proposition 3.4.0.3. $\square$

Corollary 3.4.1.3. Suppose we are given a commutative diagram of simplicial sets

3.40
\begin{equation} \label{diagram:homotopy-pullback-square5} \begin{gathered} \xymatrix { Y \ar [r] \ar [d]^{g} & X \ar [d]^{f} \\ T \ar [r] & S, } \end{gathered}\end{equation}

where $f$ is a weak homotopy equivalence. Then (3.40) is a homotopy pullback square if and only if $g$ is a weak homotopy equivalence.

Example 3.4.1.4. Suppose we are given a commutative diagram of simplicial sets

3.41
\begin{equation} \label{diagram:homotopy-pullback-square2} \begin{gathered} \xymatrix { Y \ar [r] \ar [d]^{g} & X \ar [d]^{f} \\ T \ar [r] & S, } \end{gathered}\end{equation}

where $f$ and $g$ are Kan fibrations. Then (3.41) is a homotopy pullback square if and only if, for each vertex $t \in T$ having image $s \in S$, the induced map $Y_{t} \rightarrow X_{s}$ is a weak homotopy equivalence. This is essentially a reformulation of Proposition 3.3.7.1 (by virtue of Proposition 3.4.1.2).

Example 3.4.1.5. Suppose we are given a commutative diagram of simplicial sets

3.42
\begin{equation} \label{diagram:homotopy-pullback-square3} \begin{gathered} \xymatrix { Y \ar [r] \ar [d] & X \ar [d]^{f} \\ T \ar [r] & S, } \end{gathered}\end{equation}

where $f$ is a Kan fibration. Then (3.42) is a homotopy pullback square if and only if the induced map $Y \rightarrow T \times _{S} X$ is a weak homotopy equivalence. In particular, if (3.42) is a pullback diagram, then it is also a homotopy pullback diagram. Beware that this conclusion is generally false if $f$ is not a Kan fibration.

Warning 3.4.1.6. For a general pair of morphisms $f: X \rightarrow S$, $u: T \rightarrow S$ in the category of simplicial sets, there need not exist a homotopy pullback square

\[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d]^{f} \\ T \ar [r]^{u} & S. } \]

For example, if $f: \{ 0\} \hookrightarrow \Delta ^1$ and $u: \{ 1\} \hookrightarrow \Delta ^1$ are the inclusion maps, then the existence of a commutative diagram

3.43
\begin{equation} \label{diagram:homotopy-pullback-square4} \begin{gathered} \xymatrix { Y \ar [r] \ar [d] & \{ 0\} \ar [d]^{f} \\ \{ 1\} \ar [r]^{u} & \Delta ^1 } \end{gathered}\end{equation}

guarantees that the simplicial set $Y$ is empty, in which case (3.43) is not a homotopy pullback square.

Note that Definition 3.4.1.1 is a priori asymmetric: it involves replacing the map $f: X \rightarrow S$ by a Kan fibration, but leaving the map $T \rightarrow S$ unchanged. However, this turns out to be irrelevant.

Proposition 3.4.1.7 (Symmetry). A commutative diagram of simplicial sets

\[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d]^{f} \\ T \ar [r]^{g} & S } \]

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

\[ \xymatrix { Y \ar [r] \ar [d] & T \ar [d]^{g} \\ X \ar [r]^{f} & S } \]

is a homotopy pullback square.

Proof. Using Proposition 3.1.6.1, we can choose factorizations

\[ X \xrightarrow {w_ X} X' \xrightarrow {f'} S \quad \quad T \xrightarrow {w_ T} T' \xrightarrow {g'} S \]

of $f$ and $g$, where both $f'$ and $g'$ are Kan fibrations and both $w_ X$ and $w_ T$ are weak homotopy equivalences. We can then form a commutative diagram of simplicial sets

\[ \xymatrix { Y \ar [r]^{u} \ar [d]^{v} & T' \times _{S} X \ar [r] \ar [d]^{v'} & X \ar [d]^{w_ X} \\ T \times _{S} X' \ar [r]^{u'} \ar [d] & T' \times _{S} X' \ar [r] \ar [d] & X' \ar [d]^{f'} \\ T \ar [r]^{w_ T} & T' \ar [r]^{ g'} & S. } \]

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 3.1.5.16), since the morphisms $u'$ and $v'$ are weak homotopy equivalences by virtue of Corollary 3.3.7.2. $\square$

Remark 3.4.1.8. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { Y \ar [d]^{w_ Y} \ar [r] & X \ar [d]^{w_ X} \\ Y' \ar [r] \ar [d] & X' \ar [d] \\ T \ar [r] & S, } \]

where $w_{Y}$ and $w_{X}$ are weak homotopy equivalences. Then the lower half of the diagram is a homotopy pullback square if and only if the outer rectangle is a homotopy pullback square (see Corollary 3.4.1.10 for a slight generalization).

Proposition 3.4.1.9 (Transitivity). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { Z \ar [r] \ar [d]^{h} & Y \ar [r] \ar [d]^{g} & X \ar [d]^{f} \\ U \ar [r] & T \ar [r] & S } \]

where the right half of the diagram is a homotopy pullback square. Then the left half of the diagram is a homotopy pullback square if and only if the outer rectangle is a homotopy pullback square.

Proof. By virtue of Proposition 3.1.6.1, 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. Using Proposition 3.1.6.1 again, we can factor the induced map $Y \rightarrow T \times _{S} X'$ as a composition $Y \xrightarrow { w_{Y} } Y' \xrightarrow {\overline{g}} T \times _{S} X'$, where $\overline{g}$ is a Kan fibration and $w_{Y}$ is a weak homotopy equivalence. Repeating this argument, we can factor the induced map $Z \rightarrow U \times _{T} Y'$ as a composition $Z \xrightarrow { w_{Z} } Z' \xrightarrow {\overline{h}} U \times _{T} Y'$, where $\overline{h}$ is a Kan fibration and $w_{Z}$ is a weak homotopy equivalence. We then obtain a commutative diagram

\[ \xymatrix { Z \ar [r] \ar [d]^{w_ Z} & Y \ar [r] \ar [d]^{w_ Y} & X \ar [d]^{w_ X} \\ Z' \ar [r] \ar [d]^{h'} & Y' \ar [r] \ar [d]^{g'} & X' \ar [d]^{f'} \\ U \ar [r] & T \ar [r] & S } \]

where the upper vertical maps are weak homotopy equivalences and the lower vertical maps are Kan fibrations. It follows from our assumption (and Remark 3.4.1.8) that the lower right square is a homotopy pullback. To complete the proof, it will suffice (again using Remark 3.4.1.8) to show that the lower left square is a homotopy pullback if and only if the lower rectangle is a homotopy pullback. By virtue of the criterion of Example 3.4.1.4, we are reduced to showing that for each vertex $u \in U$ having images $t \in T$ and $s \in S$, respectively, the induced map $Z'_{u} \rightarrow Y'_{t}$ is a homotopy equivalence of Kan complexes if and only if the composite map $Z'_{u} \rightarrow Y'_{t} \rightarrow X'_{s}$ is a homotopy equivalence of Kan complexes. This follows from the two-out-of-three property (Remark 3.1.5.7), since the map of fibers $Y'_{t} \rightarrow X'_{s}$ is a weak homotopy equivalence (by the criterion of Example 3.4.1.4). $\square$

Corollary 3.4.1.10 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt{ Y \ar [rr] \ar [dd] \ar [dr]^-{w_ Y} & & X \ar [dd] \ar [dr]^-{ w_{X} } & \\ & Y' \ar [rr] \ar [dd] & & X' \ar [dd] \\ T \ar [rr] \ar [dr]^-{ w_ T} & & S \ar [dr]^-{w_ S} & \\ & T' \ar [rr] & & S', } \]

where the morphisms $w_{X}$, $w_{T}$, and $w_{S}$ are weak homotopy equivalences. Then any two of the following conditions imply the third:

$(1)$

The commutative diagram

\[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d] \\ T \ar [r] & S } \]

is a homotopy pullback square.

$(2)$

The commutative diagram

\[ \xymatrix { Y' \ar [r] \ar [d] & X' \ar [d] \\ T' \ar [r] & S' } \]

is a homotopy pullback square.

$(3)$

The morphism $w_{Y}$ is a weak homotopy equivalence.

Proof. By virtue of Corollary 3.4.1.3, the bottom square in the commutative diagram

\[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d] \\ T \ar [r] \ar [d]^{w_ T} & S \ar [d]^{w_ S} \\ T' \ar [r] & S', } \]

is a homotopy pullback square. Applying Propositions 3.4.1.9 and 3.4.1.7, we see that $(1)$ is equivalent to the following:

  • The diagram

    \[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d] \\ T' \ar [r] & S' } \]

    is a homotopy pullback square.

If condition $(3)$ is satisfied, then the equivalence $(1') \Leftrightarrow (2)$ is a special case of Remark 3.4.1.8. Conversely, if $(1')$ and $(2)$ are satisfied, then Propositions 3.4.1.9 and 3.4.1.7 guarantee that the upper half of the commutative diagram

\[ \xymatrix { Y \ar [r] \ar [d]^{w_ Y} & X \ar [d]^{w_ X} \\ Y' \ar [r] \ar [d] & X \ar [d] \\ T' \ar [r] & S' } \]

is a homotopy pullback square, so that $w_{Y}$ is a weak homotopy equivalence by virtue of Corollary 3.4.1.3. $\square$

Proposition 3.4.1.11 (Summands). Suppose we are given a homotopy pullback square of simplicial sets

\[ \xymatrix { Y \ar [r]^{u} \ar [d]^{g} & X \ar [d]^{f} \\ T \ar [r]^{v} & S. } \]

Let $X' \subseteq X$, $T' \subseteq T$, and $S' \subseteq S$ be summands satisfying $f(X') \subseteq S' \supseteq v(T')$, and set $Y' = g^{-1}(T') \cap u^{-1}(X') \subseteq Y$. Then the diagram of simplicial sets

\[ \xymatrix { Y' \ar [r]^{u} \ar [d]^{g} & X' \ar [d]^{f} \\ T' \ar [r]^{v} & S' } \]

is also a homotopy pullback square.

Proof. Consider the diagram of simplicial sets

\[ \xymatrix { g^{-1}(T') \ar [r] \ar [d] & Y \ar [r] \ar [d] & X \ar [d] \\ T' \ar [r] & T \ar [r] & S. } \]

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 (Example 3.4.1.5). The square on the right is a homotopy pullbacn by assumption. Applying Proposition 3.4.1.9, we deduce that bottom half of the commutative diagram

\[ \xymatrix { Y' \ar [r] \ar [d] & X' \ar [d] \\ g^{-1}(T') \ar [r] \ar [d] & X \ar [d] \\ T' \ar [r] & S } \]

is homotopy 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 3.4.1.5). Applying Proposition 3.4.1.9 again, we conclude that the outer rectangle in the diagram

\[ \xymatrix { Y' \ar [r] \ar [d] & X' \ar@ {=} \ar [d] & X' \ar [d] \\ T' \ar [r] & S' \ar [r] & S } \]

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 3.4.1.9 again, we conclude that the left square is a homotopy pullback, as desired. $\square$