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4.5.3 Categorical Pushout Diagrams

Recall that a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l] \\ \operatorname{Fun}(C, X) \ar [u] & \operatorname{Fun}(D, X) \ar [u] \ar [l] } \]

is homotopy Cartesian (Definition 3.4.2.1). In this section, we study a stronger version of this condition.

Definition 4.5.3.1. A commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is a categorical pushout diagram if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(B,\operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}(C, \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(D, \operatorname{\mathcal{C}})^{\simeq } \ar [u] \ar [l] } \]

is homotopy Cartesian.

Remark 4.5.3.2. Every categorical pushout diagram of simplicial sets is also a homotopy pushout diagram of simplicial sets (since every Kan complex $X$ is an $\infty $-category which satisfies $\operatorname{Fun}(K, X)^{\simeq } = \operatorname{Fun}(K, X)$ for every simplicial set $X$).

We now summarize some of the basic formal properties of Definition 4.5.3.1 (which are counterparts of the analogous assertions for the class of homotopy pushout diagrams, established in ยง3.4.2).

Proposition 4.5.3.3 (Symmetry). A commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is a categorical pushout diagram if and only if the transposed diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & C \ar [d] \\ B \ar [r] & D } \]

is a categorical pushout diagram.

Proof. Apply Proposition 3.4.1.7. $\square$

Proposition 4.5.3.4 (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 square is a categorical pushout diagram. Then the right square is a categorical pushout diagram if and only if the outer rectangle is a categorical pushout diagram.

Proof. Apply Proposition 3.4.1.9. $\square$

Proposition 4.5.3.5 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt{ A \ar [rr] \ar [dd] \ar [dr]^-{w_ A} & & B \ar [dd] \ar [dr]^-{ w_{B} } & \\ & A' \ar [rr] \ar [dd] & & B' \ar [dd] \\ C \ar [rr] \ar [dr]^-{ w_ C} & & D \ar [dr]^-{w_ D} & \\ & C' \ar [rr] & & D', } \]

where the morphisms $w_{A}$, $w_{B}$, and $w_ C$ are categorical equivalences. Then any two of the following three conditions imply the third:

$(1)$

The commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is a categorical pushout square.

$(2)$

The commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ A' \ar [r] \ar [d] & B' \ar [d] \\ C' \ar [r] & D' } \]

is a categorical pushout square.

$(3)$

The morphism $w_{D}$ is a categorical equivalence.

Proposition 4.5.3.6. Suppose we are given a commutative diagram of simplicial sets

4.16
\begin{equation} \label{diagram:categorical-pushout-square0} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r]^-{f'} & D } \end{gathered} \end{equation}

where $f$ is a categorical equivalence. Then (4.16) is a categorical pushout diagram if and only if $f'$ is a categorical equivalence.

Proof. For every $\infty $-category $\operatorname{\mathcal{C}}$, we obtain a commutative diagram of simplicial sets

4.17
\begin{equation} \label{diagram:categorical-pushout-lemma} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(B,\operatorname{\mathcal{C}})^{\simeq } \ar [l]_{u} \\ \operatorname{Fun}(C,\operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(D,\operatorname{\mathcal{C}})^{\simeq }, \ar [l]_{u'} \ar [u] } \end{gathered} \end{equation}

where $u$ is a homotopy equivalence of Kan complexes (Proposition 4.5.2.8). Applying Corollary 3.4.1.3, we conclude that (4.17) is homotopy Cartesian if and only if $u$ is a homotopy equivalence of Kan complexes. Consequently, (4.16) is a categorical pushout square if and only if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(D, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(C, \operatorname{\mathcal{C}})^{\simeq }$. By virtue of Proposition 4.5.2.8, this is equivalent to the requirement that $f'$ is a weak homotopy equivalence. $\square$

Proposition 4.5.3.7. Suppose we are given a commutative diagram of simplicial sets

4.18
\begin{equation} \label{diagram:categorical-pushout-square1} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D, } \end{gathered} \end{equation}

where $f$ is a monomorphism. Then (3.46) is a categorical pushout square if and only if the induced map $\rho : C \coprod _{A} B \rightarrow D$ is a categorical equivalence of simplicial sets.

Proof. For every $\infty $-category $\operatorname{\mathcal{C}}$, we obtain a commutative diagram of Kan complexes

4.19
\begin{equation} \label{diagram:categorical-pushout-lemma2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(B,\operatorname{\mathcal{C}})^{\simeq } \ar [l]_{u} \\ \operatorname{Fun}(C,\operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(D,\operatorname{\mathcal{C}})^{\simeq }, \ar [l] \ar [u] } \end{gathered} \end{equation}

where $u$ is a Kan fibration (Corollary 4.4.5.4). It follows that the diagram (4.19) is homotopy Cartesian if and only if the induced map

\[ \theta : \operatorname{Fun}(D, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(C,\operatorname{\mathcal{C}})^{\simeq } \times _{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq }} \operatorname{Fun}(B,\operatorname{\mathcal{C}})^{\simeq } \]

is a weak homotopy equivalence (Example 3.4.1.5). By virtue of Corollary 4.4.4.6, we can identify $\theta $ with the map $\operatorname{Fun}(D, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( B \coprod _{A} C, \operatorname{\mathcal{C}})^{\simeq }$ given by precomposition with $\rho $. It follows that the diagram (3.46) is a categorical pushout square if and only if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the induced map $\operatorname{Fun}(D, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( C \coprod _{A} B, \operatorname{\mathcal{C}})^{\simeq }$ is a homotopy equivalence of Kan complexes. By virtue of Proposition 4.5.2.8, this is equivalent to the requirement that $\rho $ is a categorical equivalence. $\square$

Example 4.5.3.8. Suppose we are given a pushout diagram of simplicial sets

4.20
\begin{equation} \label{diagram:categorical-pushout-square9} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D. } \end{gathered} \end{equation}

If $f$ is a monomorphism, then (4.20) is also a homotopy pushout diagram.

Remark 4.5.3.9. Suppose we are given a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{g} & B \ar [d]^{g'} \\ C \ar [r] & D, } \]

where $f$ is a monomorphism. If $g$ is a categorical equivalence, then $g'$ is also a categorical equivalence. This follows by combining Example 4.5.3.8 with Proposition 4.5.3.6.

Corollary 4.5.3.10. Let $i: A \rightarrow B$ and $i': A' \rightarrow B'$ be morphisms of simplicial sets. Assume that $i$ is monomorphism, and that either $i$ or $i'$ is a categorical equivalence. Then the induced map

\[ (A \times B') \coprod _{ (A \times A') } (B \times A') \rightarrow B \times B' \]

is a categorical equivalence.

Proof. By virtue of Proposition 4.5.3.7, it will suffice to show that the diagram

4.21
\begin{equation} \begin{gathered}\label{diagram:categorical-pushout-corollary} \xymatrix@R =50pt@C=50pt{ A \times A' \ar [r] \ar [d] & B \times A' \ar [d] \\ A \times B' \ar [r] & B \times B' } \end{gathered} \end{equation}

is a categorical pushout square. This follows from the criterion of Proposition 4.5.3.6: if $i$ is a categorical equivalence, then the horizontal maps in the diagram (4.21) are categorical equivalences (Remark 4.5.2.7). Similarly, if $i'$ is a categorical equivalence, then the vertical maps in the diagram (4.21) are categorical equivalences. $\square$

Remark 4.5.3.11. Suppose we are given a commutative diagram of simplicial sets

4.22
\begin{equation} \label{diagram:categorical-pushout-square2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D. } \end{gathered} \end{equation}

Using Exercise 3.1.6.11, we can factor $f$ as a composition $A \xrightarrow {f'} B' \xrightarrow {w} B$, where $f'$ is a monomorphism and $w$ is a trivial Kan fibration (and therefore also a categorical equivalence, by virtue of Proposition 4.5.2.9). Combining Propositions 4.5.3.7 and 4.5.3.5, we conclude that diagram (4.22) is a categorical pushout square if and only if the induced map $u: C \coprod _{A} B' \rightarrow D$ is a categorical equivalence In particular, the condition that $u$ is a weak homotopy equivalence does not depend on the choice of factorization $f = w \circ f'$.