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3.4.2 Homotopy Pushout Squares

We now formulate a dual version of Definition 3.4.1.1.

Definition 3.4.2.1. A commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is a homotopy pushout square if, for every Kan complex $X$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l] \\ \operatorname{Fun}(C, X) \ar [u] & \operatorname{Fun}(D, X) \ar [u] \ar [l] } \]

is homotopy pullback square (Definition 3.4.1.1).

We now summarize some of the formal properties enjoyed by Definition 3.4.2.1. For the most part, these are immediate consequences of their counterparts for homotopy pullback squares (proven in ยง3.4.1).

Proposition 3.4.2.2 (Symmetry). A commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

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

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & C \ar [d] \\ B \ar [r] & D } \]

is a homotopy pushout square.

Proof. Apply Proposition 3.4.1.7. $\square$

Proposition 3.4.2.3 (Transitivity). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [r] \ar [d] & C \ar [d] \\ A' \ar [r] & B' \ar [r] & C', } \]

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

Proof. Apply Proposition 3.4.1.9. $\square$

Proposition 3.4.2.4 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt{ A \ar [rr] \ar [dd] \ar [dr]^-{w_ A} & & B \ar [dd] \ar [dr]^-{ w_{B} } & \\ & A' \ar [rr] \ar [dd] & & B' \ar [dd] \\ C \ar [rr] \ar [dr]^-{ w_ C} & & D \ar [dr]^-{w_ D} & \\ & C' \ar [rr] & & D', } \]

where the morphisms $w_{A}$, $w_{B}$, and $w_ C$ are weak homotopy equivalences. Then any two of the following three conditions imply the third:

$(1)$

The commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is a homotopy pushout square.

$(2)$

The commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ A' \ar [r] \ar [d] & B' \ar [d] \\ C' \ar [r] & D' } \]

is a homotopy pushout square.

$(3)$

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

Proposition 3.4.2.5. Suppose we are given a commutative diagram of simplicial sets

3.44
\begin{equation} \label{diagram:homotopy-pushout-square0} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r]^-{f'} & D } \end{gathered} \end{equation}

where $f$ is a weak homotopy equivalence. Then (3.44) is a homotopy pushout square if and only if $f'$ is a weak homotopy equivalence.

Proof. For every Kan complex $X$, we obtain a commutative diagram of simplicial sets

3.45
\begin{equation} \label{diagram:pushout-lemma} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l]_{u} \\ \operatorname{Fun}(C,X) \ar [u] & \operatorname{Fun}(D,X), \ar [l]_{u'} \ar [u] } \end{gathered} \end{equation}

where $u$ is a homotopy equivalence of Kan complexes (Corollary 3.1.7.5). Applying Corollary 3.4.1.3, we conclude that (3.45) is a homotopy pullback square if and only if $u$ is a homotopy equivalence of Kan complexes. Consequently, (3.44) is homotopy pushout square if and only if, for every Kan complex $X$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(D, X) \rightarrow \operatorname{Fun}(C, X)$. By virtue of Corollary 3.1.7.5, this is equivalent to the requirement that $f'$ is a weak homotopy equivalence. $\square$

Proposition 3.4.2.6. Suppose we are given a commutative diagram of simplicial sets

3.46
\begin{equation} \label{diagram:homotopy-pushout-square1} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D, } \end{gathered} \end{equation}

where $f$ is a monomorphism. Then (3.46) is a homotopy pushout square if and only if the induced map $C \coprod _{A} B \rightarrow D$ is a weak homotopy equivalence.

Proof. For every Kan complex $X$, we obtain a commutative diagram of simplicial sets

3.47
\begin{equation} \label{diagram:pushout-lemma2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l]_{u} \\ \operatorname{Fun}(C,X) \ar [u] & \operatorname{Fun}(D,X), \ar [l] \ar [u] } \end{gathered} \end{equation}

where $u$ is a Kan fibration (Corollary 3.1.3.3). It follows that the diagram (3.47) is a homotopy pullback square if and only if the induced map

\[ \operatorname{Fun}(D, X) \rightarrow \operatorname{Fun}(C,X) \times _{ \operatorname{Fun}(A,X)} \operatorname{Fun}(B,X) \simeq \operatorname{Fun}( C \coprod _{A} B, X) \]

is a weak homotopy equivalence (Example 3.4.1.5). Consequently, the diagram (3.46) is a homotopy pushout square if and only if, for every Kan complex $X$, the induced map $\operatorname{Fun}(D, X) \rightarrow \operatorname{Fun}( C \coprod _{A} B, X)$ is a homotopy equivalence of Kan complexes. By virtue of Corollary 3.1.7.5), this is equivalent to the requirement that the morphism $C \coprod _{A} B \rightarrow D$ is a weak homotopy equivalence. $\square$

Example 3.4.2.7. Suppose we are given a pushout diagram of simplicial sets

3.48
\begin{equation} \label{diagram:homotopy-pushout-square9} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D. } \end{gathered} \end{equation}

If $f$ is a monomorphism, then (3.48) is also a homotopy pushout diagram.

Remark 3.4.2.8. Suppose we are given a commutative diagram of simplicial sets

3.49
\begin{equation} \label{diagram:homotopy-pushout-square2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D. } \end{gathered} \end{equation}

Using Exercise 3.1.7.11, we can factor $f$ as a composition $A \xrightarrow {f'} B' \xrightarrow {w} B$, where $f'$ is a monomorphism and $w$ is a weak homotopy equivalence (in fact, we can even arrange that $w$ is a trivial Kan fibration). Combining Propositions 3.4.2.6 and 3.4.2.4, we conclude that diagram (3.49) is a homotopy pushout square if and only if the induced map $u: C \coprod _{A} B' \rightarrow D$ is a weak homotopy equivalence. In particular, the condition that $u$ is a weak homotopy equivalence does not depend on the choice of factorization $f = w \circ f'$.