We now formulate a dual version of Definition 3.4.1.1.
Definition 3.4.2.1. A commutative diagram of simplicial sets is a homotopy pushout square if, for every Kan complex $X$, the diagram of Kan complexes is homotopy pullback square (Definition 3.4.1.1).
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A_{01}, X) \ar [r] \ar [d] & \operatorname{Fun}(A_{0}, X) \ar [d] \\ \operatorname{Fun}(A_1, X) \ar [r] & \operatorname{Fun}(A, X) } \]
We begin by observing that if a diagram of simplicial sets
is a homotopy pushout square, then we can recover the simplicial set $A_{01}$ (up to weak homotopy equivalence) from the morphisms $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$. To see this, it will be convenient to introduce a dual version of Construction 3.4.0.3.
Construction 3.4.2.2 (Homotopy Pushouts). Let $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$ be morphisms of simplicial sets. We let $A_0 {\coprod }_{A}^{\mathrm{h}} A_{1}$ denote the iterated pushout We will refer to $A_{0} {\coprod }_{A}^{\mathrm{h}} A_{1}$ as the homotopy pushout of $A_0$ with $A_1$ along $A$. Note that the projection map $\Delta ^1 \times A \twoheadrightarrow A$ induces a comparison map $A_0 {\coprod }_{A}^{\mathrm{h}} A_{1} \twoheadrightarrow A_0 {\coprod }_{A} A_1$ from the homotopy pushout to the usual pushout, which is an epimorphism of simplicial sets.
\[ A_0 \coprod _{(\{ 0\} \times A)} (\Delta ^1 \times A) \coprod _{ (\{ 1\} \times A) } A_{1}. \]
Remark 3.4.2.3. Let $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$ be morphisms of simplicial sets, and let $X$ be a Kan complex. Then the simplicial set $\operatorname{Fun}(A, X)$ is a Kan complex (Corollary 3.1.3.4), and we have a canonical isomorphism where the right hand side is the homotopy fiber product of Construction 3.4.0.3.
\[ \operatorname{Fun}( A_{0} {\coprod }_{A}^{\mathrm{h}} A_{1}, X) \simeq \operatorname{Fun}( A_0, X) \times _{\operatorname{Fun}(A,X)}^{\mathrm{h}} \operatorname{Fun}(A_1, X), \]
Remark 3.4.2.4. Let $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$ be morphisms of simplicial sets. Then we have a canonical isomorphism $( A_0 {\coprod }_{A}^{\mathrm{h}} A_1 )^{\operatorname{op}} \simeq A_{1}^{\operatorname{op}} {\coprod }^{\mathrm{h}}_{A^{\operatorname{op}}} A_{0}^{\operatorname{op}}$.
Proposition 3.4.2.5. A commutative diagram of simplicial sets is a homotopy pushout square if and only if the induced map is a weak homotopy equivalence of simplicial sets.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]
\[ \theta : A_0 { \coprod }_{A}^{\mathrm{h}} A_{1} \twoheadrightarrow A_{0} {\coprod }_{A} A_{1} \rightarrow A_{01} \]
Proof. Let $X$ be a Kan complex, so that $\operatorname{Fun}(A,X)$ is also a Kan complex (Corollary 3.1.3.4). Applying Corollary 3.4.1.6, we see that the diagram
is a homotopy pullback square if and only if the composite map
is a homotopy equivalence. Using the isomorphism of Remark 3.4.2.3, we can identify $\rho _{X}$ with the morphism $\operatorname{Fun}( A_{01}, X) \rightarrow \operatorname{Fun}( A_{0} {\coprod }_{A}^{\mathrm{h}} A_1, X)$ given by precomposition with $\theta $. Proposition 3.4.2.5 now follows by allowing the Kan complex $X$ to vary. $\square$
We now summarize some of the formal properties enjoyed by Definition 3.4.2.1 and Construction 3.4.2.2.
Proposition 3.4.2.6. A commutative diagram of simplicial sets is a homotopy pushout square if and only if the induced diagram of opposite simplicial sets is a homotopy pushout square.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]
\[ \xymatrix@R =50pt@C=50pt{ A^{\operatorname{op}} \ar [r] \ar [d] & A_{0}^{\operatorname{op}} \ar [d] \\ A_1^{\operatorname{op}} \ar [r] & A_{01}^{\operatorname{op}} } \]
Proof. Apply Remark 3.4.1.7. $\square$
Proposition 3.4.2.7 (Symmetry). A commutative diagram of simplicial sets is a homotopy pushout square if and only if the transposed diagram is a homotopy pushout square.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_1 \ar [d] \\ A_0 \ar [r] & A_{01} } \]
Proof. Apply Proposition 3.4.1.9. $\square$
Proposition 3.4.2.8 (Transitivity). Suppose we are given a commutative diagram of simplicial sets 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.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [r] \ar [d] & C \ar [d] \\ A' \ar [r] & B' \ar [r] & C', } \]
Proof. Apply Proposition 3.4.1.11. $\square$
Proposition 3.4.2.9 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets where the morphisms $w$, $w_{0}$, and $w_{1}$ are weak homotopy equivalences. Then any two of the following three conditions imply the third:
The back face
is a homotopy pushout square.
The front face
is a homotopy pushout square.
The morphism $w_{01}$ is a weak homotopy equivalence.
\[ \xymatrix@C =50pt{ A \ar [rr] \ar [dd] \ar [dr]^-{w} & & A_{0} \ar [dd] \ar [dr]^-{ w_0 } & \\ & B \ar [rr] \ar [dd] & & B_0 \ar [dd] \\ A_{1} \ar [rr] \ar [dr]^-{ w_1} & & A_{01} \ar [dr]^-{w_{01}} & \\ & B_1 \ar [rr] & & B_{01}, } \]
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]
\[ \xymatrix@R =50pt@C=50pt{ B \ar [r] \ar [d] & B_0 \ar [d] \\ B_1 \ar [r] & B_{01} } \]
Proof. Combine Corollary 3.4.1.12 with Proposition 3.1.6.17. $\square$
Proposition 3.4.2.10. Suppose we are given a commutative diagram of simplicial sets where $f$ is a weak homotopy equivalence. Then (3.46) is a homotopy pushout square if and only if $f'$ is a weak homotopy equivalence.
3.46
\begin{equation} \label{diagram:homotopy-pushout-square0} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ A' \ar [r]^-{f'} & B' } \end{gathered} \end{equation}
Proof. For every Kan complex $X$, we obtain a commutative diagram of simplicial sets
where $u$ is a homotopy equivalence of Kan complexes (Proposition 3.1.6.17). Applying Corollary 3.4.1.5, we conclude that (3.47) is a homotopy pullback square if and only if $u$ is a homotopy equivalence of Kan complexes. Consequently, (3.46) is a homotopy pushout square if and only if, for every Kan complex $X$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(B', X) \rightarrow \operatorname{Fun}(A', X)$. By virtue of Proposition 3.1.6.17, this is equivalent to the requirement that $f'$ is a weak homotopy equivalence. $\square$
Proposition 3.4.2.11. Suppose we are given a commutative diagram of simplicial sets where $f_0$ is a monomorphism. Then (3.48) is a homotopy pushout square if and only if the induced map $A_0 \coprod _{A} A_1 \rightarrow A_{01}$ is a weak homotopy equivalence.
3.48
\begin{equation} \label{diagram:homotopy-pushout-square1} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f_0} \ar [d] & A_{0} \ar [d] \\ A_{1} \ar [r] & A_{01}, } \end{gathered} \end{equation}
Proof. For every Kan complex $X$, we obtain a commutative diagram of simplicial sets
where $u$ is a Kan fibration (Corollary 3.1.3.3). It follows that the diagram (3.49) is a homotopy pullback square if and only if the induced map
is a weak homotopy equivalence (Example 3.4.1.3). Consequently, the diagram (3.48) is a homotopy pushout square if and only if, for every Kan complex $X$, the induced map $\operatorname{Fun}(A_{01}, X) \rightarrow \operatorname{Fun}( A_0 \coprod _{A} A_1, X)$ is a homotopy equivalence of Kan complexes. By virtue of Proposition 3.1.6.17, this is equivalent to the requirement that the morphism $A_0 \coprod _{A} A_1 \rightarrow A$ is a weak homotopy equivalence. $\square$
Example 3.4.2.12. Suppose we are given a pushout diagram of simplicial sets If $f_0$ is a monomorphism, then (3.50) is also a homotopy pushout diagram.
3.50
\begin{equation} \label{diagram:homotopy-pushout-square9} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f_0} \ar [d] & A_1 \ar [d] \\ A_1 \ar [r] & A_{01}. } \end{gathered} \end{equation}
Corollary 3.4.2.13. Let $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$ be morphisms of simplicial sets. If either $f_0$ or $f_1$ is a monomorphism, then the comparison map $A_0 {\coprod }_{A}^{\mathrm{h}} A_1 \twoheadrightarrow A_0 {\coprod }_{A} A_{1}$ is a weak homotopy equivalence.
Proof. Combine Example 3.4.2.12 with Proposition 3.4.2.5. $\square$
Corollary 3.4.2.14. Suppose we are given a commutative diagram of simplicial sets where $f_0$ and $g_0$ are monomorphisms and the vertical maps are weak homotopy equivalences. Then the induced map is a weak homotopy equivalence.
\[ \xymatrix@R =50pt@C=50pt{ A_0 \ar [d] & A \ar [l]_{f_0} \ar [r]^-{f_1} \ar [d] & A_1 \ar [d] \\ B_0 & B \ar [l]_{g_0} \ar [r]^-{g_1} & B_1, } \]
\[ A_0 \coprod _{A} A_1 \rightarrow B_0 \coprod _{B} B_1 \]
Proof. Combine Example 3.4.2.12 with Proposition 3.4.2.9. $\square$
Corollary 3.4.2.15. Suppose we are given a commutative diagram of simplicial sets where the vertical maps are weak homotopy equivalences. Then the induced map is also a weak homotopy equivalence.
\[ \xymatrix@R =50pt@C=50pt{ A_0 \ar [d] & A \ar [l]_{f_0} \ar [r]^-{f_1} \ar [d] & A_1 \ar [d] \\ B_0 & B \ar [l]_{g_0} \ar [r]^-{g_1} & B_1, } \]
\[ A_0 {\coprod }^{\mathrm{h}}_{A} A_1 \rightarrow B_0 {\coprod }^{\mathrm{h}}_{B} B_1 \]
Proof. Apply Corollary 3.4.2.14 to the diagram
Let us conclude with an application of these concepts.
Proposition 3.4.2.16. Suppose we are given a commutative diagram of simplicial sets with the following property: for every simplex $\sigma : \Delta ^ k \rightarrow S$, the induced map $f_{\sigma }: \Delta ^ k \times _{S} X \rightarrow \Delta ^ k \times _{S} Y$ is a weak homotopy equivalence of simplicial sets. Then $f$ is a weak homotopy equivalence of simplicial sets.
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S & } \]
Proof. We will prove the following stronger assertion: for every morphism of simplicial sets $S' \rightarrow S$, the induced map
is a weak homotopy equivalence of simplicial sets. By virtue of Proposition 3.2.8.3, (and Remark 1.1.4.4), we may assume without loss of generality that $S'$ has dimension $\leq k$ for some integer $k \geq -1$. We proceed by induction on $k$. In the case $k=-1$, the simplicial set $S'$ is empty and there is nothing to prove. Assume therefore that $k \geq 0$. Let $S''$ denote the $(k-1)$-skeleton of $S'$ and let $I$ be the set of nondegenerate $k$-simplices of $S'$, so that Proposition 1.1.4.12 supplies a pushout diagram of simplicial sets
where the horizontal maps are monomorphisms. It follows that the front and back faces of the diagram
are homotopy pushout squares (Proposition 3.4.2.11). Consequently, to show that $f_{S'}$ is a weak homotopy equivalence, it will suffice to show that $f_{S''}$, $u$, and $v$ are weak homotopy equivalences (Proposition 3.4.2.9). In the first two cases, this follows from our inductive hypothesis. We may therefore replace $S'$ by the coproduct $\coprod _{i \in I} \Delta ^{k}$, and thereby reduce to the case of a coproduct of simplices. Using Remark 3.1.6.20, we can further reduce to the case where $S' \simeq \Delta ^{k}$ is a standard simplex, in which case the desired result follows from our hypothesis on $f$. $\square$
Corollary 3.4.2.17. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Suppose that, for every $k$-simplex $\Delta ^ k \rightarrow S$, the fiber product $\Delta ^{k} \times _{S} X$ is weakly contractible. Then $f$ is a weak homotopy equivalence.
Proof. Apply Proposition 3.4.2.16 in the special case $Y = S$. $\square$