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3.4.4 Mather's First Cube Theorem

Our goal in this section is to prove a converse of Theorem 3.4.3.3, known as Mather's first cube theorem. As before, we begin with an elementary statement about the category of sets.

Exercise 3.4.4.1. Suppose we are given a commutative diagram of sets

\[ \xymatrix { \overline{C} \ar [d] & \overline{A} \ar [l] \ar [r]^-{\overline{i}} \ar [d] & \overline{B} \ar [d] \\ C & A \ar [l] \ar [r]^-{i} & B } \]

where both squares are pullback diagrams, and $i$ is a monomorphism (so that $\overline{i}$ is also a monomorphism). Show that both squares in the resulting diagram

\[ \xymatrix { \overline{C} \ar [r] \ar [d] & \overline{C} \coprod _{ \overline{A} } \overline{B} \ar [d] & \overline{B} \ar [l] \ar [d] \\ C \ar [r] & C \coprod _{A} B & B \ar [l] } \]

are pullback squares.

Warning 3.4.4.2. The conclusion of Exercise 3.4.4.1 does not necessarily hold if the map $i$ is not injective. For example, let $G$ be a group with multiplication map $m: G \times G \rightarrow G$, and let $\pi ,\pi ': G \times G \rightarrow G$ be the projection maps onto the two factors. Then the diagram of sets

\[ \xymatrix { G \ar [d] & G \times G \ar [l]_-{\pi } \ar [r]^-{\pi '} \ar [d]^{m} & G \ar [d] \\ \ast & G \ar [l] \ar [r] & \ast } \]

consists of pullback squares, but the induced diagram

\[ \xymatrix { G \ar [d] \ar [r] & G \coprod _{ G \times G} G \ar [d] & G \ar [d] \ar [l] \\ \ast \ar [r] & \ast \coprod _{G} \ast & \ast \ar [l] } \]

does not (except in the case where $G$ is trivial).

Exercise 3.4.4.1 has an analogue for homotopy pullback diagrams of simplicial sets.

Proposition 3.4.4.3. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \overline{C} \ar [d] & \overline{A} \ar [l] \ar [r]^-{\overline{i}} \ar [d] & \overline{B} \ar [d] \\ C & A \ar [r]^-{i} \ar [l] & B, } \]

in which both squares are homotopy Cartesian. If $i$ and $\overline{i}$ are monomorphisms, then both squares in the induced diagram

\[ \xymatrix { \overline{C} \ar [r] \ar [d] & \overline{C} \coprod _{ \overline{A} } \overline{B} \ar [d] & \overline{B} \ar [l] \ar [d] \\ C \ar [r] & C \coprod _{ A} B & B \ar [l] } \]

are also homotopy Cartesian.

Proposition 3.4.4.3 is an immediate consequence of Example 3.4.2.7 together with the following homotopy-invariant statement:

Theorem 3.4.4.4 (Mather's First Cube Theorem). Suppose we are given a cubical diagram of simplicial sets

3.53
\begin{equation} \label{equation:mather-first-cube1} \begin{gathered} \xymatrix {\overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd] \\ A \ar [rr] \ar [dr] & & B \ar [dr] & \\ & C \ar [rr] & & D } \end{gathered}\end{equation}

having the property that the back and left faces

\[ \xymatrix { \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] & \overline{A} \ar [r] \ar [d] & \overline{C} \ar [d] \\ A \ar [r] & B & A \ar [r] & C } \]

are homotopy Cartesian, and the top and bottom faces

\[ \xymatrix { \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] & A \ar [r] \ar [d] & B \ar [d] \\ \overline{C} \ar [r] & \overline{D} & C \ar [r] & D } \]

are homotopy coCartesian. Then the front and right faces

\[ \xymatrix { \overline{C} \ar [r] \ar [d] & \overline{D} \ar [d] & \overline{B} \ar [r] \ar [d] & \overline{D} \ar [d] \\ C \ar [r] & D & B \ar [r] & D } \]

are also homotopy Cartesian.

Proof. The proof will proceed in several steps, each of which involves replacing one or more of the terms in (3.53) by a weakly equivalent simplicial set (by virtue of Corollary 3.4.1.10 and Proposition 3.4.2.4, such replacements will not affect the truth of our hypotheses or of the desired conclusion). Let us denote each of the morphisms appearing in the diagram (3.53) by $f_{XY}$, where $X,Y \in \{ \overline{A}, \overline{B}, \overline{C}, \overline{D}, A, B, C, D \} $ are the source and target of $f_{XY}$, respectively.

  • By virtue of Proposition 3.1.7.1, the morphism $f_{\overline{B} B}: \overline{B} \rightarrow B$ factors as a composition $\overline{B} \xrightarrow {w} \overline{B}' \xrightarrow { f_{\overline{B} B}' } B$, where $w$ is anodyne and $f_{ \overline{B} B }'$ is a Kan fibration. Replacing $\overline{B}$ by $\overline{B}'$ (and $\overline{D}$ by the pushout $\overline{B}' \coprod _{ \overline{B} } \overline{D}$), we can reduce to the case where $f_{\overline{B} B}$ is a Kan fibration. Similarly, we can arrange that the map $f_{\overline{C} C}: \overline{C} \rightarrow C$ is a Kan fibration.

  • By virtue of Exercise 3.1.7.7, the morphism $f_{ \overline{A} \overline{C} }$ factors as a composition $\overline{A} \xrightarrow { f_{ \overline{A} \overline{C} }' } \overline{C}' \xrightarrow {w} \overline{C}$, where $f_{\overline{A} \overline{C}}'$ is a monomorphism and $w$ is a trivial Kan fibration. Replacing $\overline{C}$ by $\overline{C}'$, we can reduce to the case where $f_{ \overline{A} \overline{C} }$ is a monomorphism. Note that this replacement does not injure our hypothesis that $f_{\overline{C} C}$ is a Kan fibration.

  • Since the left face

    3.54
    \begin{equation} \label{equation:mather-first-cube2} \begin{gathered} \xymatrix { \overline{A} \ar [r]^-{ f_{ \overline{A} \overline{C} }} \ar [d]^{ f_{\overline{A} A}} & \overline{C} \ar [d]^{ f_{ \overline{C} C} } \\ A \ar [r] & C } \end{gathered} \end{equation}

    is homotopy Cartesian and $f_{\overline{C} C}$ is a Kan fibration, the induced map $\overline{A} \rightarrow A \times _{C} \overline{C}$ is a weak homotopy equivalence (Example 3.4.1.5). We may therefore replace $\overline{A}$ by the fiber product $A \times _{C} \overline{C}$, and thereby reduce to the case where the diagram (3.54) is a pullback square (so that $f_{ \overline{A} A}$ is also a Kan fibration). Note that this replacement does not injure our hypothesis that $f_{ \overline{A} \overline{C}}$ is a monomorphism.

  • By virtue of Exercise 3.1.7.7, the morphism $f_{AB}$ factors as a composition $A \xrightarrow { f_{AB}' } B' \xrightarrow {w} B$, where $f_{AB}'$ is a monomorphism and $w$ is a trivial Kan fibration. Replacing $B$ by $B'$ (and $\overline{B}$ by the fiber product $B' \times _{B} \overline{B}'$), we can reduce to the case where $f_{AB}$ is a monomorphism.

  • By virtue of Exercise 3.1.7.7, the morphism $f_{\overline{A} \overline{B}}$ factors as a composition $\overline{A} \xrightarrow { f_{\overline{A} \overline{B}}' } \overline{B}' \xrightarrow {w} \overline{B}$, where $f_{\overline{A} \overline{B}}'$ is a monomorphism and $w$ is a trivial Kan fibration. Replacing $\overline{B}$ by $\overline{B}'$, we can reduce to the case where $f_{ \overline{A} \overline{B} }$ is a monomorphism.

  • The back face

    3.55
    \begin{equation} \label{equation:mather-first-cube3} \begin{gathered} \xymatrix { \overline{A} \ar [r]^-{ f_{ \overline{A} \overline{B} }} \ar [d]^{ f_{\overline{A} A}} & \overline{B} \ar [d]^{ f_{ \overline{B} B} } \\ A \ar [r]^-{f_{AB}} & B } \end{gathered} \end{equation}

    is a homotopy pullback square in which the horizontal maps are monomorphisms and the vertical maps are Kan fibrations. It follows that, for every vertex $a \in A$ having image $b = f_{AB}(a) \in B$, the induced map of fibers $\overline{A}_{a} \rightarrow \overline{B}_{b}$ is a homotopy equivalence. Let $\overline{B}' \subseteq \overline{B}$ denote the simplicial subset spanned by those simplices $\sigma : \Delta ^{n} \rightarrow \overline{B}$ having the property that the restriction $\sigma |_{ A \times _{B} \Delta ^{n} }$ factors through $\overline{A}$. Applying Lemma 3.3.8.4, we deduce that the restriction $f_{ \overline{B} B}|_{ \overline{B}' }: \overline{B}' \rightarrow B$ is also a Kan fibration. Moreover, the inclusion map $\overline{B}' \hookrightarrow \overline{B}$ induces a homotopy equivalence of fibers $\overline{B}'_{b} \hookrightarrow \overline{B}_{b}$, for each vertex $b \in B$. It follows that the inclusion $\overline{B}' \hookrightarrow \overline{B}$ is a weak homotopy equivalence (Corollary 3.3.7.3). Replacing $\overline{B}$ by $\overline{B}'$, we can reduce to the case where the diagram (3.55) is a pullback square.

  • By assumption, the top and bottom faces

    3.56
    \begin{equation} \label{equation:mather-first-cube4} \begin{gathered} \xymatrix { \overline{A} \ar [r] \ar [d]^{ f_{\overline{A} \overline{C}}} & \overline{B} \ar [d] & A \ar [r] \ar [d]^{ f_{AC} } & B \ar [d] \\ \overline{C} \ar [r] & \overline{D} & C \ar [r] & D } \end{gathered} \end{equation}

    are homotopy coCartesian. Since $f_{ \overline{A} \overline{C} }$ and $f_{AC}$ are monomorphisms, it follows that the induced maps

    \[ \overline{C} \coprod _{\overline{A}} \overline{B} \rightarrow \overline{D} \quad \quad C \coprod _{A} B \rightarrow D \]

    are weak homotopy equivalences (Proposition 3.4.2.6). We may therefore replace $D$ by the pushout $C \coprod _{A} B$ and $\overline{D}$ by the pushout $\overline{C} \coprod _{\overline{A} } \overline{B}$, and thereby reduce to the case where the diagrams (3.56) are pushout squares.

  • Applying Exercise 3.4.4.1 levelwise, we deduce that the front and right faces

    3.57
    \begin{equation} \label{equation:mather-first-cube5} \begin{gathered} \xymatrix { \overline{C} \ar [r] \ar [d]^{ f_{\overline{C} C}} & \overline{D} \ar [d]^{ f_{ \overline{D} D}} & \overline{B} \ar [r] \ar [d]^{ f_{\overline{B} B}} & \overline{D} \ar [d]^{ f_{\overline{D} D}} \\ C \ar [r] & D & B \ar [r] & D } \end{gathered} \end{equation}

    are pullback squares in the category of simplicial sets. In particular, for every simplex $\sigma : \Delta ^ n \rightarrow D$, the projection map $\Delta ^ n \times _{ D} \overline{D} \rightarrow \Delta ^ n$ is a pullback either of $f_{ \overline{B} B}$ or of $f_{ \overline{C} C}$, and is therefore a Kan fibration. Applying Remark 3.1.1.6, we conclude that $f_{ \overline{D} D}: \overline{D} \rightarrow D$ is a Kan fibration. It follows that the diagrams (3.57) are also homotopy pullback squares, as desired.

$\square$