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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

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A_{01}, X) \ar [r] \ar [d] & \operatorname{Fun}(A_{0}, X) \ar [d] \\ \operatorname{Fun}(A_1, X) \ar [r] & \operatorname{Fun}(A, X) } \]

is homotopy pullback square (Definition 3.4.1.1).

We begin by observing that if a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f_0} \ar [d]^{f_{1}} & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

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

\[ A_0 \coprod _{(\{ 0\} \times A)} (\Delta ^1 \times A) \coprod _{ (\{ 1\} \times A) } A_{1}. \]

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.

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

\[ \operatorname{Fun}( A_{0} {\coprod }_{A}^{\mathrm{h}} A_{1}, X) \simeq \operatorname{Fun}( A_0, X) \times _{\operatorname{Fun}(A,X)}^{\mathrm{h}} \operatorname{Fun}(A_1, X), \]

where the right hand side is the homotopy fiber product of Construction 3.4.0.3.

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

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

is a homotopy pushout square if and only if the induced map

\[ \theta : A_0 { \coprod }_{A}^{\mathrm{h}} A_{1} \twoheadrightarrow A_{0} {\coprod }_{A} A_{1} \rightarrow A_{01} \]

is a weak homotopy equivalence of simplicial sets.

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A_{01}, X) \ar [r] \ar [d] & \operatorname{Fun}(A_0,X) \ar [d] \\ \operatorname{Fun}(A_1,X) \ar [r] & \operatorname{Fun}(A,X) } \]

is a homotopy pullback square if and only if the composite map

\[ \rho _{X}: \operatorname{Fun}( A_{01}, X) \rightarrow \operatorname{Fun}( A_0, X) \times _{ \operatorname{Fun}( A,X) } \operatorname{Fun}( A_1, X) \hookrightarrow \operatorname{Fun}( A_{0} ,X) \times ^{\mathrm{h}}_{\operatorname{Fun}(A,X)} \operatorname{Fun}(A_1,X) \]

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

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

is a homotopy pushout square if and only if the induced diagram of opposite simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A^{\operatorname{op}} \ar [r] \ar [d] & A_{0}^{\operatorname{op}} \ar [d] \\ A_1^{\operatorname{op}} \ar [r] & A_{01}^{\operatorname{op}} } \]

is a homotopy pushout square.

Proof. Apply Remark 3.4.1.7. $\square$

Proposition 3.4.2.7 (Symmetry). A commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

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

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_1 \ar [d] \\ A_0 \ar [r] & A_{01} } \]

is a homotopy pushout square.

Proof. Apply Proposition 3.4.1.9. $\square$

Proposition 3.4.2.8 (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 3.4.1.11. $\square$

Proposition 3.4.2.9 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt{ A \ar [rr] \ar [dd] \ar [dr]^-{w} & & A_{0} \ar [dd] \ar [dr]^-{ w_0 } & \\ & B \ar [rr] \ar [dd] & & B_0 \ar [dd] \\ A_{1} \ar [rr] \ar [dr]^-{ w_1} & & A_{01} \ar [dr]^-{w_{01}} & \\ & B_1 \ar [rr] & & B_{01}, } \]

where the morphisms $w$, $w_{0}$, and $w_{1}$ are weak homotopy equivalences. Then any two of the following three conditions imply the third:

$(1)$

The back face

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

is a homotopy pushout square.

$(2)$

The front face

\[ \xymatrix@R =50pt@C=50pt{ B \ar [r] \ar [d] & B_0 \ar [d] \\ B_1 \ar [r] & B_{01} } \]

is a homotopy pushout square.

$(3)$

The morphism $w_{01}$ is a weak homotopy equivalence.

Proposition 3.4.2.10. Suppose we are given a commutative diagram of simplicial sets

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}

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.

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

3.47
\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}(A',X) \ar [u] & \operatorname{Fun}(B',X), \ar [l]_{u'} \ar [u] } \end{gathered} \end{equation}

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

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}

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.

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

3.49
\begin{equation} \label{diagram:pushout-lemma2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(A_0,X) \ar [l]_{u} \\ \operatorname{Fun}(A_1,X) \ar [u] & \operatorname{Fun}(A_{01},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 (3.49) is a homotopy pullback square if and only if the induced map

\[ \operatorname{Fun}(A_{01}, X) \rightarrow \operatorname{Fun}(A_0,X) \times _{ \operatorname{Fun}(A,X)} \operatorname{Fun}(A_1,X) \simeq \operatorname{Fun}( A_0 \coprod _{A} A_1, X) \]

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

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}

If $f_0$ is a monomorphism, then (3.50) is also a homotopy pushout diagram.

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.

Corollary 3.4.2.14. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A_0 \ar [d] & A \ar [l]_{f_0} \ar [r]^-{f_1} \ar [d] & A_1 \ar [d] \\ B_0 & B \ar [l]_{g_0} \ar [r]^-{g_1} & B_1, } \]

where $f_0$ and $g_0$ are monomorphisms and the vertical maps are weak homotopy equivalences. Then the induced map

\[ A_0 \coprod _{A} A_1 \rightarrow B_0 \coprod _{B} B_1 \]

is a weak homotopy equivalence.

Corollary 3.4.2.15. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A_0 \ar [d] & A \ar [l]_{f_0} \ar [r]^-{f_1} \ar [d] & A_1 \ar [d] \\ B_0 & B \ar [l]_{g_0} \ar [r]^-{g_1} & B_1, } \]

where the vertical maps are weak homotopy equivalences. Then the induced map

\[ A_0 {\coprod }^{\mathrm{h}}_{A} A_1 \rightarrow B_0 {\coprod }^{\mathrm{h}}_{B} B_1 \]

is also a weak homotopy equivalence.

Proof. Apply Corollary 3.4.2.14 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times A \ar [d] & \operatorname{\partial \Delta }^1 \times A \ar [l] \ar [r] \ar [d] & A_0 \coprod A_1 \ar [d] \\ \Delta ^1 \times B & \operatorname{\partial \Delta }^1 \times B \ar [l] \ar [r] & B_0 \coprod B_1. } \]
$\square$

Let us conclude with an application of these concepts.

Proposition 3.4.2.16. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S & } \]

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.

Proof. We will prove the following stronger assertion: for every morphism of simplicial sets $S' \rightarrow S$, the induced map

\[ f_{S'}: S' \times _{S} X \rightarrow S' \times _{S} Y \]

is a weak homotopy equivalence of simplicial sets. By virtue of Proposition 3.2.8.3, (and Remark 1.1.3.6), 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 1.1.3.13 supplies a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { i \in I }{\coprod } \operatorname{\partial \Delta }^{k} \ar [r] \ar [d] & \underset { i \in I }{\coprod } \Delta ^{k} \ar [d] \\ S'' \ar [r] & S', } \]

where the horizontal maps are monomorphisms. It follows that the front and back faces of the diagram

\[ \xymatrix@R =50pt@C=40pt{ (\underset { i \in I }{\coprod } \operatorname{\partial \Delta }^{k}) \times _{S} X \ar [rr] \ar [dd] \ar [dr]^{u} & & \underset { i \in I }{\coprod } (\Delta ^{k} \times _{S} X) \ar [dd] \ar [dr]^{v} & \\ & \underset { i \in I }{\coprod } (\operatorname{\partial \Delta }^{k} \times _{S} Y) \ar [dd] \ar [rr] & & \underset { i \in I }{\coprod } (\Delta ^{k} \times _{S} Y) \ar [dd] \\ S'' \times _{S} X \ar [rr] \ar [dr]^{f_{S''}} & & S' \times _{S} X \ar [dr]^{ f_{S'} } & \\ & S'' \times _{S} Y \ar [rr] & & S' \times _{S} Y } \]

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 $Y = S$. $\square$