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4.5.4 Categorical Pushout Squares

Recall that 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

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

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

Definition 4.5.4.1. 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 categorical pushout square if, for every $\infty$-category $\operatorname{\mathcal{C}}$, the diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(A_0,\operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}(A_1, \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(A_{01}, \operatorname{\mathcal{C}})^{\simeq } \ar [u] \ar [l] }$

is a homotopy pullback square.

Remark 4.5.4.2. Every categorical pushout square of simplicial sets is also a homotopy pushout square 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 $K$).

Remark 4.5.4.3. Suppose we are given a categorical pushout square of simplicial sets

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

Then, for every simplicial set $K$, the induced diagram

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

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

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A \times K,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(A_0 \times K,\operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}(A_1 \times K, \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(A_{01} \times K, \operatorname{\mathcal{C}})^{\simeq } \ar [u] \ar [l] }$

is a homotopy pullback square. This follows by applying the requirement Definition 4.5.4.1 to the $\infty$-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$.

Proposition 4.5.4.4. A commutative diagram of simplicial sets

4.30
$$\begin{gathered}\label{equation:categorical-pushout-to-categorical-pullback} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \end{gathered}$$

is a categorical pushout square if and only if it satisfies the following condition:

$(\ast )$

For every $\infty$-category $\operatorname{\mathcal{C}}$, the diagram of $\infty$-categories

4.31
$$\begin{gathered}\label{equation:categorical-pushout-to-categorical-pullback2} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}}) & \operatorname{Fun}( A_0, \operatorname{\mathcal{C}}) \ar [l] \\ \operatorname{Fun}(A_1, \operatorname{\mathcal{C}}) \ar [u] & \operatorname{Fun}( A_{01}, \operatorname{\mathcal{C}}) \ar [l] \ar [u] } \end{gathered}$$

is a categorical pullback square.

Proof. Fix an $\infty$-category $\operatorname{\mathcal{C}}$. If the diagram of $\infty$-categories (4.31) is a categorical pullback square, then the diagram of cores

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}( A_0, \operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}(A_1, \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}( A_{01}, \operatorname{\mathcal{C}})^{\simeq } \ar [l] \ar [u] }$

is a homotopy pullback square (Proposition 4.5.2.12). Allowing $\operatorname{\mathcal{C}}$ to vary, we see that if $(\ast )$ is satisfied, then (4.30) is a categorical pushout square. For the converse, assume that (4.30) is a categorical pullback square. For every simplicial set $X$, the simplicial set $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$ is an $\infty$-category (Theorem 1.4.3.7), so the diagram of Kan complexes

4.32
$$\begin{gathered}\label{equation:categorical-pushout-to-categorical-pullback25} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( A, \operatorname{Fun}(X, \operatorname{\mathcal{C}}))^{\simeq } & \operatorname{Fun}(A_0, \operatorname{Fun}(X, \operatorname{\mathcal{C}}) )^{\simeq } \ar [l] \\ \operatorname{Fun}(A_1,\operatorname{Fun}( X, \operatorname{\mathcal{C}}))^{\simeq } \ar [u] & \operatorname{Fun}(A_{01}, \operatorname{Fun}(X, \operatorname{\mathcal{C}}))^{\simeq } \ar [u] \ar [l] } \end{gathered}$$

is a homotopy pullback square. Identifying (4.32) with the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( X, \operatorname{Fun}(A, \operatorname{\mathcal{C}}))^{\simeq } & \operatorname{Fun}(X, \operatorname{Fun}(A_0, \operatorname{\mathcal{C}}) )^{\simeq } \ar [l] \\ \operatorname{Fun}(X,\operatorname{Fun}( A_1, \operatorname{\mathcal{C}}))^{\simeq } \ar [u] & \operatorname{Fun}(X, \operatorname{Fun}(A_{01}, \operatorname{\mathcal{C}}))^{\simeq } \ar [u] \ar [l] }$

and allowing $X$ to vary, we conclude that the diagram (4.31) is a categorical pullback square (Proposition 4.5.2.12). $\square$

Corollary 4.5.4.5. Suppose we are given a categorical pushout square of simplicial sets

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

where the horizontal maps are monomorphisms. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. For every diagram $A' \rightarrow \operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}_{A'/}(B', \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{A/}(B,\operatorname{\mathcal{C}})$ is an equivalence of $\infty$-categories.

Proof. Proposition 4.5.4.4 guarantees that the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B', \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}(A', \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}( A, \operatorname{\mathcal{C}}) }$

is a categorical pullback square, and Corollary 4.4.5.3 guarantees that the vertical maps are isofibrations. The desired result now follows from Corollary 4.5.2.25. $\square$

Proposition 4.5.4.6. 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 categorical 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 categorical pushout square.

Proof. Apply Remark 3.4.1.7. $\square$

Proposition 4.5.4.7 (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 categorical 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 categorical pushout square.

Proof. Apply Proposition 3.4.1.9. $\square$

Proposition 4.5.4.8 (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 square is a categorical pushout. Then the right square is a categorical pushout if and only if the outer rectangle is a categorical pushout.

Proof. Apply Proposition 3.4.1.11. $\square$

Proposition 4.5.4.9 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@C =50pt{ A \ar [rr] \ar [dd] \ar [dr]^-{w} & & A_0 \ar [dd] \ar [dr]^-{ w_{0} } & \\ & B \ar [rr] \ar [dd] & & B_0 \ar [dd] \\ A_1 \ar [rr] \ar [dr]^-{ w_1} & & A_{01} \ar [dr]^-{w_{01}} & \\ & B_0 \ar [rr] & & B_{01}, }$

where the morphisms $w$, $w_{0}$, and $w_{1}$ are categorical 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 categorical 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 categorical pushout square.

$(3)$

The morphism $w_{01}$ is a categorical equivalence of simplicial sets.

Proposition 4.5.4.10. Suppose we are given a commutative diagram of simplicial sets

4.33
$$\label{diagram:categorical-pushout-square0} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & A_0 \ar [d] \\ A_1 \ar [r]^-{f'} & A_{01} } \end{gathered}$$

where $f$ is a categorical equivalence. Then (4.33) is a categorical pushout square 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

4.34
$$\label{diagram:categorical-pushout-lemma} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(A_0,\operatorname{\mathcal{C}})^{\simeq } \ar [l]_{u} \\ \operatorname{Fun}(A_1,\operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(A_{01},\operatorname{\mathcal{C}})^{\simeq }, \ar [l]_{u'} \ar [u] } \end{gathered}$$

where $u$ is a homotopy equivalence of Kan complexes (Proposition 4.5.3.8). Applying Corollary 3.4.1.5, we conclude that (4.34) is a homotopy pullback square if and only if $u'$ is a homotopy equivalence of Kan complexes. Consequently, (4.33) is a categorical pushout square if and only if, for every $\infty$-category $\operatorname{\mathcal{C}}$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(A_{01}, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(A_1, \operatorname{\mathcal{C}})^{\simeq }$. By virtue of Proposition 4.5.3.8, this is equivalent to the requirement that $f'$ is a categorical equivalence. $\square$

Proposition 4.5.4.11. Suppose we are given a commutative diagram of simplicial sets

4.35
$$\begin{gathered}\label{diagram:categorical-pushout-square1} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01}, } \end{gathered}$$

where $f$ is a monomorphism. Then (4.35) is a categorical pushout square if and only if the induced map $\rho : A_0 \coprod _{A} A_1 \rightarrow A_{01}$ is a categorical equivalence of simplicial sets.

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

4.36
$$\label{diagram:categorical-pushout-lemma2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}}) & \operatorname{Fun}(A_0,\operatorname{\mathcal{C}}) \ar [l]_{u} \\ \operatorname{Fun}(A_1,\operatorname{\mathcal{C}}) \ar [u] & \operatorname{Fun}(A_{01},\operatorname{\mathcal{C}}), \ar [l] \ar [u] } \end{gathered}$$

where $u$ is an isofibration (Corollary 4.4.5.3). It follows that the diagram (4.36) is a categorical pullback square if and only if the induced map

$\theta _{\operatorname{\mathcal{C}}}: \operatorname{Fun}(A_{01}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A_0,\operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})} \operatorname{Fun}(A_1,\operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( A_0 \coprod _{A} A_1, \operatorname{\mathcal{C}})$

is an equivalence of $\infty$-categories (Proposition 4.5.2.20). Using Proposition 4.5.4.4, we see that this condition is satisfied for every $\infty$-category $\operatorname{\mathcal{C}}$ if and only if (4.35) is a categorical pushout square. The desired result now follows from Proposition 4.5.3.8. $\square$

Example 4.5.4.12. Suppose we are given a pushout diagram of simplicial sets

4.37
$$\label{diagram:categorical-pushout-square9} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01}. } \end{gathered}$$

If $f$ is a monomorphism, then (4.37) is also a categorical pushout square.

Remark 4.5.4.13. Suppose we are given a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{g} & A_0 \ar [d]^{g'} \\ A_1 \ar [r] & A_{01}, }$

where $f$ is a monomorphism. If $g$ is a categorical equivalence, then $g'$ is also a categorical equivalence. This follows by combining Example 4.5.4.12 with Proposition 4.5.4.10.

Corollary 4.5.4.14. 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 categorical equivalences. Then the induced map

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

is a categorical equivalence.

Corollary 4.5.4.15. Let $i: A \rightarrow B$ and $i': A' \rightarrow B'$ be morphisms of simplicial sets. Assume that $i$ is a 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 4.5.4.11, it will suffice to show that the diagram

4.38
$$\begin{gathered}\label{diagram:categorical-pushout-corollary} \xymatrix@R =50pt@C=50pt{ A \times A' \ar [r] \ar [d] & B \times A' \ar [d] \\ A \times B' \ar [r] & B \times B' } \end{gathered}$$

is a categorical pushout square. This follows from the criterion of Proposition 4.5.4.10: if $i$ is a categorical equivalence, then the horizontal maps in the diagram (4.38) are categorical equivalences (Remark 4.5.3.7). Similarly, if $i'$ is a categorical equivalence, then the vertical maps in the diagram (4.38) are categorical equivalences. $\square$