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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 a diagram of simplicial sets $X_0 \rightarrow X \leftarrow X_1$, we can form the fiber product $X_0 \times _{X} X_1$. Beware that, in general, this construction is not invariant under weak homotopy equivalence:

Warning 3.4.0.1. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [d] \ar [r] & X \ar [d] & X_1 \ar [l] \ar [d] \\ Y_0 \ar [r] & Y & Y_1 \ar [l] } \]

for which the vertical maps are weak homotopy equivalences. Then the induced map

\[ X_0 \times _{X} X_1 \rightarrow Y_0 \times _{Y} Y_1 \]

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 $\Delta ^0$.

Under some mild assumptions, the bad behavior described in Warning 3.4.0.1 can be avoided.

Proposition 3.4.0.2. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [d] \ar [r]^-{f} & X \ar [d] & X_1 \ar [l] \ar [d] \\ Y_0 \ar [r]^-{f'} & Y & Y_1, \ar [l] } \]

where the vertical maps are weak homotopy equivalences. If $f$ and $f'$ are Kan fibrations, then the induced map $X_0 \times _{X} X_1 \rightarrow Y_0 \times _{Y} Y_1$ is a weak homotopy equivalence.

Proof. We have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X_0 \times _{X} X_1 \ar [r] \ar [d] & Y_0 \times _{Y} Y_1 \ar [d] \\ X_1 \ar [r] & Y_1, } \]

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 $x \in X_1$ having image $y \in Y_1$, the induced map of fibers

\[ (X_0 \times _{X} X_1)_{x} \simeq X_0 \times _{X} \{ x\} \rightarrow Y_0 \times _{Y} \{ y\} = (Y_0 \times _{Y} Y_1)_{y} \]

is a homotopy equivalence of Kan complexes. This follows by applying Proposition 3.3.7.1 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [d]^{f} \ar [r] & Y_0 \ar [d]^{f'} \\ X \ar [r] & Y. } \]
$\square$

To address the phenomenon described in Warning 3.4.0.1 more generally, it is convenient to work with a homotopy-invariant replacement for the fiber product.

Construction 3.4.0.3 (The Homotopy Fiber Product). Let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be morphisms of simplicial sets, where $X$ is a Kan complex. We let $X_0 \times _{X}^{\mathrm{h}} X_1$ denote the simplicial set

\[ X_0 \times _{ \operatorname{Fun}( \{ 0\} , X) } \operatorname{Fun}( \Delta ^1, X ) \times _{ \operatorname{Fun}( \{ 1\} , X) } X_1. \]

We will refer to $X_0 \times _{X}^{\mathrm{h}} X_1$ as the homotopy fiber product of $X_0$ with $X_1$ over $X$.

Warning 3.4.0.4. For any diagram of simplicial sets $X_0 \rightarrow X \leftarrow X_1$, the simplicial set $X_0 \times _{ \operatorname{Fun}( \{ 0\} , X) } \operatorname{Fun}( \Delta ^1, X ) \times _{ \operatorname{Fun}( \{ 1\} , X) } X_1$ is well-defined. However, we will refer to it as a homotopy fiber product (and denote it by $X_0 \times ^{\mathrm{h}}_{X} X_1$) only in the case where $X$ is a Kan complex. In more general situations, we will refer to this simplicial set as the oriented fiber product of $X_0$ with $X_1$ over $X$, and denote it by $X_0 \operatorname{\vec{\times }}_{X} X_1$ (Definition 4.6.4.1). In the setting of $\infty $-categories, we will adopt a slightly different definition for the homotopy fiber product $X_0 \times _{X}^{\mathrm{h}} X_1$: see Construction 4.5.2.1.

Example 3.4.0.5. Let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be continuous functions between topological spaces. We let $X_0 \times ^{\mathrm{h}}_{X} X_1$ denote the set of all triples $(x_0, x_1, p)$ where $x_0$ is a point of $X_0$, $x_1$ is a point of $X_1$, and $p: [0,1] \rightarrow X$ is a continuous function satisfying $p(0) = f_0(x_0)$ and $p(1) = f_1(x_1)$. We will refer to $X_0 \times ^{\mathrm{h}}_{X} X_1$ as the homotopy fiber product of $X_0$ with $X_1$ over $X$. The homotopy fiber product $X_1 \times ^{\mathrm{h}}_{X} X_1$ carries a natural topology, given by viewing it as a subspace of the product $X_0 \times X_1 \times \operatorname{Hom}_{\operatorname{Top}}( [0,1], X)$ (where we endow the path space $\operatorname{Hom}_{\operatorname{Top}}([0,1],X)$ with the compact-open topology). We then have a canonical isomorphism of simplicial sets

\[ \operatorname{Sing}_{\bullet }( X_0 \times _{X}^{\mathrm{h}} X_1) \simeq \operatorname{Sing}_{\bullet }(X_0) \times ^{\mathrm{h}}_{ \operatorname{Sing}_{\bullet }(X) } \operatorname{Sing}_{\bullet }(X_1) \]

where the right hand side is the homotopy fiber product of Kan complexes given in Construction 3.4.0.3.

Remark 3.4.0.6 (Homotopy Fibers). Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then $f$ is a homotopy equivalence if and only if, for each vertex $y \in Y$, the homotopy fiber $X \times ^{\mathrm{h}}_{Y} \{ y\} $ is a contractible Kan complex. To see this, we observe that $f$ factors as a composition

\[ X \xrightarrow {\delta } X \times _{Y}^{\mathrm{h}} Y \xrightarrow {\pi } Y, \]

where $\delta $ is a homotopy equivalence and $\pi $ is a Kan fibration (Example 3.1.7.10). It follows that $f$ is a homotopy equivalence if and only if $\pi $ is a homotopy equivalence, which is equivalent to the requirement that each fiber $\pi ^{-1} \{ y\} = X \times _{Y}^{\mathrm{h}} \{ y\} $ is contractible (Proposition 3.3.7.4).

In the situation of Construction 3.4.0.3, the diagonal inclusion

\[ X \hookrightarrow \operatorname{Fun}(\Delta ^1, X) \quad \quad x \mapsto \operatorname{id}_ x \]

induces a monomorphism from the ordinary fiber product $X_0 \times _{X} X_1$ to the homotopy fiber product $X_0 \times _{X}^{\mathrm{h}} X_1$.

Proposition 3.4.0.7. Let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be morphisms of simplicial sets. Assume that $X$ is a Kan complex and that either $f_0$ or $f_1$ is a Kan fibration. Then the inclusion map $X_0 \times _{X} X_1 \rightarrow X_{0} \times _{X}^{\mathrm{h}} X_1$ is a weak homotopy equivalence.

Proof. Without loss of generality we may assume that $f_0$ is a Kan fibration. Since $X$ is a Kan complex, the evaluation map $\operatorname{ev}_0: \operatorname{Fun}(\Delta ^1, X) \rightarrow \operatorname{Fun}(\{ 1\} , X)$ is a trivial Kan fibration (Corollary 3.1.3.6), and therefore induces a trivial Kan fibration $q: \operatorname{Fun}(\Delta ^1,X) \times _{ \operatorname{Fun}(\{ 1\} , X) } X_1 \rightarrow X_1$. The diagonal map $\delta : X \hookrightarrow \operatorname{Fun}(\Delta ^1,X)$ determines a map $s: X_1 \rightarrow \operatorname{Fun}(\Delta ^1,X) \times _{ \operatorname{Fun}(\{ 1\} , X) } X_1$ which is a section of $q$, and therefore also a weak homotopy equivalence. The desired result now follows by applying Proposition 3.4.0.2 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar@ {=}[d] \ar [r]^-{f_0} & X \ar@ {=}[d] & X_1 \ar [l]_{f_1} \ar [d]^{s} \\ X_0 \ar [r]^-{f_0} & X & \operatorname{Fun}( \Delta ^1, X) \times _{ \operatorname{Fun}(\{ 1\} , X) } X_1. \ar [l] } \]
$\square$

Warning 3.4.0.8. The conclusion of Proposition 3.4.0.7 is generally false if neither $f_0$ or $f_1$ is assumed to be a Kan fibration. For example, suppose that $X$ is a Kan complex containing vertices $x$ and $y$. If $x \neq y$, then the fiber product $\{ x\} \times _{X} \{ y\} $ is empty. However, the homotopy fiber product $\{ x\} \times ^{\mathrm{h}}_{X} \{ y\} $ is not necessarily empty: its vertices can be identified with edges $p: x \rightarrow y$ having source $x$ and target $y$.

In general, the failure of the inclusion map $X_0 \times _{X} X_1 \hookrightarrow X_0 \times ^{\mathrm{h}}_{X} X_1$ to be a weak homotopy equivalence should be viewed as a feature, rather than a bug. From the perspective of homotopy theory, the homotopy fiber product is better behaved than the ordinary fiber product:

Proposition 3.4.0.9. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [r] \ar [d] & X \ar [d] & X_1 \ar [l] \ar [d] \\ Y_0 \ar [r] & Y & Y_1 \ar [l] } \]

where $X$ and $Y$ are Kan complexes and the vertical maps are weak homotopy equivalences. Then the induced map $X_0 \times _{X}^{\mathrm{h}} X_1 \rightarrow Y_0 \times _{Y}^{\mathrm{h}} Y_1$ is also a weak homotopy equivalence.

Proof. Apply Proposition 3.4.0.2 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^1, X) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, X) \ar [d] & X_0 \times X_1 \ar [l] \ar [d] \\ \operatorname{Fun}( \Delta ^1, Y) \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, Y) & Y_0 \times Y_1, \ar [l] } \]

noting that the left horizontal maps are Kan fibrations by virtue of Corollary 3.1.3.3. $\square$

Warning 3.4.0.10. Let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be morphisms of simplicial sets, where $X$ is a Kan complex. The homotopy fiber products $X_0 \times ^{\mathrm{h}}_{X} X_1$ and $X_{1} \times _{X}^{\mathrm{h}} X_0$ are generally not isomorphic as simplicial sets. Instead, we have a canonical isomorphism

\[ (X_1 \times ^{\mathrm{h}}_{X} X_0)^{\operatorname{op}} \simeq X_{0}^{\operatorname{op}} \times ^{\mathrm{h}}_{ X^{\operatorname{op}} } X_1^{\operatorname{op}}. \]

However, $X_0 \times ^{\mathrm{h}}_{X} X_1$ and $X_{1} \times _{X}^{\mathrm{h}} X_0$ have the same weak homotopy type. To see this, we can use Proposition 3.1.7.1 to factor $f_0$ as a composition $X_0 \xrightarrow {w} X'_0 \xrightarrow {f'_0} X$, where $w$ is a weak homotopy equivalence and $f'_{0}$ is a Kan fibration. Using Propositions 3.4.0.7 and 3.4.0.9, we see that the maps

\[ X_0 \times ^{\mathrm{h}}_{X} X_1 \rightarrow X'_{0} \times _{X}^{\mathrm{h}} X_1 \hookleftarrow X'_0 \times _{X} X_1 \simeq X_1 \times _{X} X'_0 \hookrightarrow X_1 \times _{X}^{\mathrm{h}} X'_{0} \leftarrow X_1 \times _{X}^{\mathrm{h}} X_0 \]

are weak homotopy equivalences.

For many applications, it will be useful to reformulate the notion of homotopy fiber product by viewing it as a property of diagrams, rather than as a construction. Recall that a commutative diagram of simplicial sets

3.36
\begin{equation} \begin{gathered}\label{equation:homotopy-pullback-candidate} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \end{gathered} \end{equation}

is a pullback square if the induced map $\theta : X_{01} \rightarrow X_{0} \times _{X} X_1$ is an isomorphism of simplicial sets. If $X$ is a Kan complex, we will say that the diagram (3.36) is a homotopy pullback square if the composite map

\[ X_{01} \xrightarrow {\theta } X_{0} \times _{X} X_{1} \hookrightarrow X_{0} \times _{X}^{\mathrm{h}} X_1 \]

is a weak homotopy equivalence of simplicial sets. In §3.4.1, we give an overview of the theory of homotopy pullback diagrams (beginning with an extension to the case where $X$ is not a Kan complex: see Definition 3.4.1.1 and Corollary 3.4.1.6).

The preceding discussion has an analogue for pushout diagrams. Given morphisms of simplicial sets $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$ having the same source, we define the homotopy pushout of $A_0$ with $A_1$ along $A$ to be the iterated coproduct

\[ A_{0} {\coprod }^{\mathrm{h}}_{A} A_1 = A_0 \coprod _{ (\{ 0 \} \times A)} ( \Delta ^1 \times A) \coprod _{( \{ 1\} \times A) } A_1 \]

(Construction 3.4.2.2). We say that a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt@R=50pt{ A \ar [r]^-{f_0} \ar [d]^{f_1} & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

is a homotopy pushout square if the induced map

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

is a weak homotopy equivalence (Proposition 3.4.2.5). Many of the basic properties of homotopy pullback diagrams have counterparts for homotopy pushout diagrams, which we summarize in §3.4.2.

The notions of homotopy pullback and homotopy pushout diagram were 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 Seifert-van Kampen theorem (Theorem 3.4.7.1) and the excision theorem for singular homology (Theorem 3.4.7.3).

Remark 3.4.0.11. The notions of homotopy pullback and homotopy pushout diagrams can be regarded as homotopy-invariant replacements for the usual notion of pullback and pushout diagrams, respectively. We will later make this heuristic precise by showing that a commutative diagram in the ordinary category of Kan complexes

\[ \xymatrix@C =50pt@R=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \]

is a homotopy pullback square (homotopy pushout square) if and only if it is a pullback square (pushout square) when regarded as a diagram in the $\infty $-category $\operatorname{\mathcal{S}}$ of Kan complexes (Construction 5.5.1.1); see Examples 7.6.4.2 and 7.6.4.3.

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 van-Kampen Theorem