3.4 Homotopy Pullback and Homotopy Pushout Squares
Recall that the category of simplicial sets admits arbitrary limits and colimits (Remark 1.1.1.13). In particular, given diagrams of simplicial sets
\[ T \rightarrow S \leftarrow X \quad \quad \quad \quad B \leftarrow A \rightarrow C, \]
we can form the pullback $T \times _{S} X$ and the pushout $B \coprod _{A} C$. Beware that, in general, neither of these constructions is invariant under weak homotopy equivalence.
Warning 3.4.0.1. Suppose we are given a commutative diagram of of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ T \ar [d] \ar [r] & S \ar [d] & X \ar [l] \ar [d] \\ T' \ar [r] & S' & X' \ar [l] } \]
for which the vertical maps are weak homotopy equivalences. Then the induced map $T \times _{S} X \rightarrow T' \times _{S'} X'$ need not be a weak homotopy equivalence. For example, the pullback of the upper half of the diagram
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [r] \ar@ {=}[d] & \Delta ^1 \ar [d] & \{ 1\} \ar@ {=}[d] \ar [l] \\ \{ 0\} \ar [r]^{\sim } & \Delta ^0 & \{ 1\} , \ar [l]_{\sim } } \]
is empty, while the pullback of the lower half is isomorphic to $\Delta ^0$.
Warning 3.4.0.2. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ C \ar [d] & A \ar [d] \ar [l] \ar [r] & B \ar [d] \\ C' & A' \ar [l] \ar [r] & B' } \]
in which the vertical maps are weak homotopy equivalences. Then the induced map $C \coprod _{A} B \rightarrow C' \coprod _{A'} B'$ need not be a weak homotopy equivalence. For example, the pushout of the upper half of the diagram
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \ar [d] & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] \ar@ {=}[d] & \Delta ^1 \ar [d] \\ \Delta ^0 & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] & \Delta ^0 } \]
is not weakly contractible (it has nontrivial homology in degree $1$), but the pushout of the lower half is isomorphic to $\Delta ^0$.
One can avoid the troublesome phenomena of Warnings 3.4.0.1 and 3.4.0.2 by restricting attention to diagrams which satisfy some additional conditions.
Proposition 3.4.0.3. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ T \ar [d] \ar [r] & S \ar [d] & X \ar [l]_{f} \ar [d] \\ T' \ar [r] & S' & X', \ar [l]_{f'} } \]
where the vertical maps are weak homotopy equivalences. If $f$ and $f'$ are Kan fibrations, then the induced map $T \times _{S} X \rightarrow T' \times _{S'} X'$ is a weak homotopy equivalence.
Proof.
We have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ T \times _{S} X \ar [r] \ar [d] & T' \times _{S'} X' \ar [d] \\ T \ar [r] & T', } \]
where the vertical maps are Kan fibrations (since they are pullbacks of $f$ and $f'$, respectively). By virtue of Proposition 3.3.7.1, it will suffice to show that for each vertex $t \in T$ having image $t' \in T'$, the induced map of fibers
\[ (T \times _{S} X)_{t} \simeq \{ t\} \times _{S} X \rightarrow \{ t'\} \times _{S'} X' = (T' \times _{S'} X')_{t'} \]
is a homotopy equivalence of Kan complexes. This follows from the criterion of Proposition 3.3.7.1, applied to the diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r] & X' \ar [d]^{f'} \\ S \ar [r] & S'. } \]
$\square$
Proposition 3.4.0.4. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ C \ar [d] & A \ar [d] \ar [l] \ar [r]^{i} & B \ar [d] \\ C' & A' \ar [l] \ar [r]^{i'} & B', } \]
where the vertical maps are weak homotopy equivalences. If $i$ and $i'$ are monomorphisms, then the induced map $\theta : C \coprod _{A} B \rightarrow C' \coprod _{A'} B'$ is also a weak homotopy equivalence.
Proof.
Let $X$ be a Kan complex; we will show that composition with $\theta $ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}( C' \coprod _{A'} B', X) \rightarrow \operatorname{Fun}( C \coprod _{A} B, X)$ (and therefore a bijection on connected components). This follows by applying Proposition 3.4.0.3 to the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(C',X) \ar [r] \ar [d] & \operatorname{Fun}(A', X) \ar [d] & \operatorname{Fun}(B', X) \ar [l] \ar [d] \\ \operatorname{Fun}(C, X) \ar [r] & \operatorname{Fun}(A, X) & \operatorname{Fun}(B, X) \ar [l]; } \]
note that the vertical maps in this diagram are weak homotopy equivalences by virtue of Corollary 3.1.7.5, and the right horizontal maps are Kan fibrations by virtue of Corollary 3.1.3.3
$\square$
We can summarize Proposition 3.4.0.3 and Warning 3.4.0.1 more informally as asserting that the pullback construction
\[ ( T \rightarrow S \xleftarrow {f} X ) \mapsto T \times _{S} X \]
has good behavior when $f$ is a Kan fibration of simplicial sets, but not in general. Fortunately, the assumption that $f$ is a Kan fibration is easy to arrange, using the fibrant replacement functor of Proposition 3.1.7.1.
Construction 3.4.0.5 (The Homotopy Fiber Product). Suppose we are given a diagram of simplicial sets $T \rightarrow S \xleftarrow {f} X$. By virtue of Proposition 3.1.7.1, the morphism $f$ factors as a composition
\[ X \xrightarrow {w} X' \xrightarrow {f'} S, \]
where $f'$ is a Kan fibration and $w$ is a weak homotopy equivalence. We will refer to the fiber product $T \times _{S} X'$ as a homotopy fiber product of $T$ with $X$ (relative to $S$), and denote it by $T \times _{S}^{h} X$.
Exercise 3.4.0.6 (Invariance of the Homotopy Fiber Product). Let $T \rightarrow S \xleftarrow {f} X$ be a diagram of simplicial sets. Show that, up to weak homotopy equivalence, the homotopy fiber product $T \times _{S}^{h} X$ does not depend on the factorization of $f$ chosen in Construction 3.4.0.5. In other words, if $f$ admits a pair of 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, then then the simplicial sets $T \times _{S} X'$ and $T \times _{S} X''$ have the same weak homotopy type (hint: reduce to the case where $w'$ is anodyne, and use Proposition 3.4.0.3). See Proposition 3.4.1.2 for a related statement.
Exercise 3.4.0.7 (Symmetry). Let $T \rightarrow S \leftarrow X$ be a diagram of simplicial sets. Show that the homotopy fiber products $T \times _{S}^{h} X$ and $X \times _{S}^{h} T$ have the same weak homotopy type (see Proposition 3.4.1.7 for a related statement).
Example 3.4.0.8 (Path Spaces). Let $S$ be a Kan complex containing a pair of vertices $s,t \in S$, and let
\[ P_{s,t} = \{ s\} \times _{ \operatorname{Fun}( \{ 0\} , S) } \operatorname{Fun}( \Delta ^1, S) \times _{ \operatorname{Fun}(\{ 1\} , S) } \{ t\} \]
denote the Kan complex parametrizing paths in $S$ from $s$ to $t$ (so that the vertices of $P_{s,t}$ can be identified with edges of $S$ which originate at the vertex $s$ and terminate at the vertex $t$). Then $P_{s,t}$ can be identified with the fiber product $\{ s\} \times _{S} Q$, where $Q = \operatorname{Fun}( \Delta ^1, S) \times _{ \operatorname{Fun}( \{ 1\} , S) } \{ t\} $ is a contractible Kan complex and the evaluation map $Q \rightarrow S$ is a Kan fibration (see Example 3.1.7.10). It follows that the Kan complex $P_{s,t}$ is a homotopy fiber product $\{ s\} \times _{S}^{h} \{ t\} $.
An awkward feature of Construction 3.4.0.5 is that, although the homotopy fiber product $T \times _{S}^{h} X$ is welldefined up to weak homotopy equivalence (Exercise 3.4.0.6), its isomorphism class depends on an auxiliary choice (namely, a factorization of the map $f: X \rightarrow S$ as a composition $f' \circ w$, where $f'$ is a Kan fibration and $w$ a weak homotopy equivalence). We will address this point in §3.4.1 by adopting a different perspective. Suppose we are given a commutative diagram of simplicial sets
3.36
\begin{equation} \label{diagram:homotopypullbacksquareA} \begin{gathered} \xymatrix@R =50pt@C=50pt{ Y \ar [r] \ar [d] & X \ar [d] \\ T \ar [r] & S. } \end{gathered}\end{equation}
We will say that (3.36) is a homotopy pullback square if the composite map
\[ Y \rightarrow T \times _{S} X \rightarrow T \times _{S}^{h} X \]
is a weak homotopy equivalence (Definition 3.4.1.1). It is not hard to see that this condition does not depend on the factorization chosen in the construction of $T \times _{S}^{h} X$ (see Proposition 3.4.1.2, which is essentially a more precise version of Exercise 3.4.0.6).
Construction 3.4.0.5 has a counterpart for pushout squares. Given a diagram of simplicial sets $C \leftarrow A \xrightarrow {i} B$, we can always factor $i$ as a composition $A \xrightarrow {i'} B' \xrightarrow {w} B$, where $i'$ is a monomorphism and $w$ is a weak homotopy equivalence (Exercise 3.1.7.11). In this case, we denote the pushout $C \coprod _{A} B'$ by $C \coprod _{A}^{h} B$ and refer to it as the homotopy pushout of $C$ and $B$ along $A$. Using Proposition 3.4.0.4, it is not difficult to see that the weak homotopy type of $C \coprod _{A}^{h} B$ is independent of the choice of factorization $i = w \circ i'$. However, the isomorphism class of $C \coprod _{A}^{h} B$ does depend on this choice, which makes the construction somewhat cumbersome to work with. We remedy this point in §3.4.2 by introducing a collection of commutative diagrams
3.37
\begin{equation} \label{diagram:homotopypushoutsquareA} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D} \end{gathered}\end{equation}
which we will refer to as homotopy pushout squares (Definition 3.4.2.1), which are characterized by the requirement that the composite map $C \coprod ^{h}_{A} B \rightarrow C \coprod _{A} B \rightarrow D$ is a weak homotopy equivalence (Remark 3.4.2.8).
The theory of homotopy pullback and homotopy pushout diagrams was introduced by Mather (in the setting of topological spaces, rather than simplicial sets) and have subsequently proven to be a very useful tool in algebraic topology. In [MR402694], Mather established two fundamental results relating homotopy pullback and homotopy pushout squares, which are now known as the Mather cube theorems:
Suppose we are given a homotopy pushout square of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D. } \]
If $\overline{D} \rightarrow D$ is a Kan fibration, then the induced diagram
\[ \xymatrix@R =50pt@C=50pt{ A \times _{D} \overline{D} \ar [r] \ar [d] & B \times _{D} \overline{D} \ar [d] \\ C \times _{D} \overline{D} \ar [r] & \overline{D} } \]
is also a homotopy pushout square (Proposition 3.4.3.2). Stated more informally, the collection of homotopy pushout squares is stable under pullback by Kan fibrations. In §3.4.3 we establish a slightly more general (and homotopy invariant) version of this statement, which is known as Mather's second cube theorem (Theorem 3.4.3.3).
Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \overline{C} \ar [d] & \overline{A} \ar [l] \ar [r]^{\overline{i}} \ar [d] & \overline{B} \ar [d] \\ C & A \ar [r]^{i} \ar [l] & B, } \]
in which both squares are homotopy pullbacks. If $i$ and $\overline{i}$ are monomorphisms, then both squares in the induced diagram
\[ \xymatrix@R =50pt@C=50pt{ \overline{C} \ar [r] \ar [d] & \overline{C} \coprod _{ \overline{A} } \overline{B} \ar [d] & \overline{B} \ar [l] \ar [d] \\ C \ar [r] & C \coprod _{ A} B & B \ar [l] } \]
are also homotopy pullback squares (Proposition 3.4.4.3). In §3.4.4 we establish a slightly more general (and homotopy invariant) version of this statement, which is known as Mather's first cube theorem (Theorem 3.4.4.4).
The homotopy theory of topological spaces provides a rich supply of examples of homotopy pushout squares. Let $X$ be a topological space which can be written as the union of two open subsets $U,V \subseteq X$. In §3.4.6, we show that the resulting diagram of singular simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sing}_{\bullet }(U \cap V) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(U) \ar [d] \\ \operatorname{Sing}_{\bullet }(V) \ar [r] & \operatorname{Sing}_{\bullet }(X) } \]
is a homotopy pushout square (Theorem 3.4.6.1). To carry out the proof, we make use of the fact that the weak homotopy type of a simplicial set $X$ can be recovered from its underlying semisimplicial set (see Proposition 3.4.5.4 and Corollary 3.4.5.5), which we explain in §3.4.5). We conclude in §3.4.7 by applying Theorem 3.4.6.1 to deduce the classical Seifertvan Kampen theorem (Theorem 3.4.7.1) and the excision theorem for singular homology (Theorem 3.4.7.3).
Structure

Subsection 3.4.1: Homotopy Pullback Squares

Subsection 3.4.2: Homotopy Pushout Squares

Subsection 3.4.3: Mather's Second Cube Theorem

Subsection 3.4.4: Mather's First Cube Theorem

Subsection 3.4.5: Digression: Weak Homotopy Equivalences of Semisimplicial Sets

Subsection 3.4.6: Excision

Subsection 3.4.7: The Seifert vanKampen Theorem