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7.6.3 Pullbacks and Pushouts

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Recall that a commutative square in $\operatorname{\mathcal{C}}$ is a morphism of simplicial sets $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ which we represent informally by a diagram

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d] & Y' \ar [d] \\ X \ar [r] & Y } \]

(see Example 1.4.2.15). Note that the simplicial set $\Delta ^1 \times \Delta ^1 \simeq \operatorname{N}_{\bullet }( [1] \times [1] )$ can be regarded both as a left cone (on the nerve of the partially ordered set $[1] \times [1] \setminus \{ (0,0) \} $) and as a right cone (on the nerve of the partially ordered set $[1] \times [1] \setminus \{ (1,1) \} $).

Definition 7.6.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square. We say that $\sigma $ is a pullback square if it is a limit diagram in $\operatorname{\mathcal{C}}$ (see Definition 7.1.2.4), and that $\sigma $ is a pushout square if it is a colimit diagram in $\operatorname{\mathcal{C}}$.

Example 7.6.3.2. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then diagram $\sigma : [1] \times [1] \rightarrow \operatorname{\mathcal{C}}$ is a pullback square in $\operatorname{\mathcal{C}}$ (in the sense of classical category theory) if and only if the induced map

\[ \operatorname{N}_{\bullet }(\sigma ): \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \]

is a pullback square in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 7.6.3.1); this follows from Example 7.1.1.4 and Remark 7.1.2.6. Similarly, $\sigma $ is a pushout square in $\operatorname{\mathcal{C}}$ if and only if $\operatorname{N}_{\bullet }(\sigma )$ is a pushout square in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Warning 7.6.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a morphism, which we depict as a diagram

\[ \xymatrix@C =50pt@R=50pt{ X_{01} \ar [r]^-{g_0} \ar [d]^-{g_1} & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]

Beware that, if $\sigma $ is a pullback square in the $\infty $-category $\operatorname{\mathcal{C}}$, then the associated diagram

\[ \xymatrix@C =50pt@R=50pt{ X_{01} \ar [r]^-{[g_0]} \ar [d]^-{[g_1]} & X_0 \ar [d]^-{[f_0]} \\ X_1 \ar [r]^-{[f_1]} & X } \]

need not be a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (see Example 7.6.3.4 and Exercise 7.6.3.5). If $Y$ is an object of $\operatorname{\mathcal{C}}$, then the map of sets

\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X_{01} ) \xrightarrow { ( [g_0], [g_1]) \circ } \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X_0) \times _{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X ) } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y, X_1 ) \]

is surjective, but need not be injective. Given a commutative diagram

7.56
\begin{equation} \begin{gathered}\label{equation:pullback-not-preserved-by-homotopy} \xymatrix@C =50pt@R=50pt{ Y \ar [r]^-{[g_0]} \ar [d]^-{[g_1]} & X_0 \ar [d]^-{[f_0]} \\ X_1 \ar [r]^-{[f_1]} & X } \end{gathered} \end{equation}

in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, we can always find a morphism $g_{01}: Y \rightarrow X_{01}$ satisfying $g_{01}: Y \rightarrow X_{01}$ satisfying $[g_0] = [f'_0] \circ [g_{01}]$ and $[ g_1 ] = [ f'_1 ] \circ [ g_{01} ]$. However, the homotopy class $[g_{01}]$ is not uniquely determined: roughly speaking, to construct $g_{01}$, we need to lift (7.56) to a commutative diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Such a lift always exists (Exercise 1.4.2.10), but is not unique (even up to homotopy).

Example 7.6.3.4. Let $q: X \rightarrow S$ be a Kan fibration between Kan complexes, let $s \in S$ be a vertex, and let $X_{s}$ denote the fiber $\{ s\} \times _{S} X$. Then the commutative diagram of simplicial sets

7.57
\begin{equation} \begin{gathered}\label{equation:disagreement-of-fiber-products} \xymatrix@R =50pt@C=50pt{ X_{s} \ar [r] \ar [d] & X \ar [d]^-{q} \\ \{ s\} \ar [r] & S } \end{gathered} \end{equation}

is a homotopy pullback square (Example 3.4.1.3), and therefore induces a pullback square in the $\infty $-category $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})$ (see Example 7.6.4.2). However, if $X$ is contractible and $X_{s}$ is not, then (7.57) is not a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Exercise 7.6.3.5. Let $G$ be a group and let $H \subseteq G$ be a commutative normal subgroup, so that we have a commutative diagram of Kan complexes

7.58
\begin{equation} \begin{gathered}\label{equation:exact-sequence-pullback-square} \xymatrix@C =50pt@R=50pt{ B_{\bullet }H \ar [r] \ar [d] & B_{\bullet } G \ar [d] \\ \Delta ^0 \ar [r] & B_{\bullet }(G/H) } \end{gathered} \end{equation}

  • Show that (7.58) is a pullback diagram in the ordinary category of Kan complexes, and that it determines a pullback diagram in the $\infty $-category $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})$ (see Example 7.6.4.2).

  • Show that, if $H$ is contained in the center of $G$, then the diagram (7.58) is also pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

  • Show that, if $H$ is not contained in the center of $G$, then the diagram $B_{\bullet } G \rightarrow B_{\bullet } (G/H) \leftarrow \Delta ^0$ does not have a limit in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. In particular, the diagram (7.58) is not a pullback square in $\mathrm{h} \mathit{\operatorname{Kan}}$.

Remark 7.6.3.6 (Symmetry). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square in $\operatorname{\mathcal{C}}$, and let $\sigma ': \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ denote the commutative square which is obtained from $\sigma $ by precomposing with the automorphism of $\Delta ^1 \times \Delta ^1$ given by permuting the factors. Then $\sigma $ is a pullback square if and only if $\sigma '$ is a pullback square, and $\sigma $ is a pushout square if and only if $\sigma '$ is a pushout square.

Remark 7.6.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative diagram in $\operatorname{\mathcal{C}}$. Then $\sigma $ is a pushout diagram in $\operatorname{\mathcal{C}}$ if and only if the opposite diagram $\sigma ^{\operatorname{op}}: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a pullback diagram in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$; here we implicitly identify the simplicial set $\Delta ^1 \times \Delta ^1$ with its opposite (beware that there are two possible identifications we could choose, but the choice does not matter by virtue of Remark 7.6.3.6).

Remark 7.6.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma ,\sigma ': \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be square diagrams which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}})$). Then $\sigma $ is a pullback square if and only if $\sigma '$ is a pullback square, and a pushout square if and only if $\sigma '$ is a pushout square.

Notation 7.6.3.9 (Fiber Products). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and suppose we are given a pair of morphisms $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ of $\operatorname{\mathcal{C}}$ having the same target. It follows from Proposition 7.1.1.12 that if there exists a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X } \]

in $\operatorname{\mathcal{C}}$, then the object $X_{01}$ is determined up to isomorphism by $f_0$ and $f_1$. To emphasize this, we will often denote the object $X_{01}$ by $X_0 \times _{X} X_1$ and refer to it as the fiber product of $X_0$ with $X_1$ over $X$. Similarly, if there exists a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ Y \ar [r]^-{g_0} \ar [d]^-{g_1} & Y_0 \ar [d] \\ Y_1 \ar [r] & Y_{01} } \]

in $\operatorname{\mathcal{C}}$, then the object $Y_{01}$ is determined up to isomorphism by $g_0$ and $g_1$. To emphasize this, we often denote the object $Y_{01}$ by $Y_{0} \coprod _{ Y} Y_1$ and refer to it as the pushout of $Y_0$ with $Y_1$ along $Y$.

Definition 7.6.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will say that $\operatorname{\mathcal{C}}$ admits pullbacks if, for every pair of morphisms $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ having the same target, there exists a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]

We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves pullbacks if, for every pullback square $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the composition $(F \circ \sigma ): \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{D}}$ is a pullback square in the $\infty $-category $\operatorname{\mathcal{D}}$.

We say that $\operatorname{\mathcal{C}}$ admits pushouts if, for every pair of morphisms $g_0: Y \rightarrow Y_0$ and $g_1: Y \rightarrow Y_1$ having the same source, there exists a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ Y \ar [r]^-{g_0} \ar [d]^-{g_1} & Y_0 \ar [d] \\ Y_1 \ar [r] & Y_{01}. } \]

We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves pullbacks if, for every pushout square $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the composition $(F \circ \sigma ): \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{D}}$ is a pushout square in the $\infty $-category $\operatorname{\mathcal{D}}$.

Proposition 7.6.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square, which we represent by a diagram

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

Then $\sigma $ is a pullback diagram in $\operatorname{\mathcal{C}}$ if and only if it exhibits $X_{01}$ as a product of $X_0$ with $X_1$ in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$.

Proof. This is a special case of Remark 7.1.2.11. $\square$

Stated more informally, a fiber product $X_0 \times _{ X } X_1$ (formed in an $\infty $-category $\operatorname{\mathcal{C}}$) is a product of $X_0$ with $X_1$ in the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$.

Corollary 7.6.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits pullbacks if and only if, for each object $X \in \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ admits finite products.

Proof. By virtue of Proposition 7.6.3.11, the $\infty $-category $\operatorname{\mathcal{C}}$ admits pullbacks if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ admits pairwise products. Since $\operatorname{\mathcal{C}}_{/X}$ has an initial object (given by the identity morphism $\operatorname{id}_{X}: X \rightarrow X$; see Proposition 4.6.6.23), this is equivalent to the requirement that $\operatorname{\mathcal{C}}_{/X}$ admits finite products (Corollary 7.6.1.20). $\square$

Remark 7.6.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories, where $\operatorname{\mathcal{C}}$ admits pullbacks. Then $F$ preserves pullbacks if and only if, for each object $X \in \operatorname{\mathcal{C}}$, the induced functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{D}}_{ / F(X) }$ preserves finite products.

Corollary 7.6.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square, which we represent by a diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f_0} \ar [d]^-{f_1} & X_0 \ar [d] \\ X_1 \ar [r] & \mathbf{1}. } \]

Suppose that ${\bf 1}$ is a final object of $\operatorname{\mathcal{C}}$. Then $\sigma $ is a pullback square if and only if the morphisms $f_0$ and $f_1$ exhibit $X$ as a coproduct of $X_0$ with $X_1$ in the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof. The assumption that ${\bf 1}$ is final guarantees that the projection map $\operatorname{\mathcal{C}}_{ / {\bf 1} } \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration (Proposition 4.6.6.11), so that the desired result follows from the criterion of Proposition 7.6.3.11. $\square$

Proposition 7.6.3.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square, represented informally by the diagram

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

Then:

$(1)$

If $f$ is an isomorphism, then $\sigma $ is a pullback square if and only if $f'$ is also an isomorphism.

$(2)$

If $f'$ is an isomorphism, then $\sigma $ is a pushout square if and only if $f$ is also an isomorphism.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Note that $\sigma $ restricts to a diagram

\[ \sigma _0: \operatorname{N}_{\bullet }( [1] \times [1] \setminus \{ (0,0) \} ) \rightarrow \operatorname{\mathcal{C}} \]

satisfying $\sigma _0( 0,1) = X$, $\sigma _0( 1,1) = Y$, and $\sigma _0( 1,0 ) = Y'$. The assumption that $f$ is an isomorphism guarantees that $\sigma _0$ is right Kan extended from the full subcategory

\[ \{ 1\} \times \Delta ^1 \subseteq \operatorname{N}_{\bullet }([1] \times [1] \setminus \{ (0,0) \} ). \]

It follows that $\sigma $ is a pullback diagram if and only if the restriction $\sigma |_{ \operatorname{N}_{\bullet }( \{ (0,0) < (1,0) < (1,1) \} )}$ is a limit diagram (Corollary 7.3.7.2) By virtue of Corollary 7.2.2.6, this is equivalent to the requirement that $f'$ is an isomorphism. $\square$

Proposition 7.6.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^2 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be diagram, which we depict informally as

7.59
\begin{equation} \begin{gathered}\label{equation:transitivity-pullback-square} \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d] & Y' \ar [r] \ar [d] & Z' \ar [d] \\ X \ar [r] & Y \ar [r] & Z. } \end{gathered} \end{equation}

Then:

$(1)$

Assume that the right square of (7.59) is a pullback. Then the left square is a pullback if and only if the outer rectangle is a pullback.

$(2)$

Assume that the left square of (7.59) is a pushout. Then the left square is a pushout if and only if the outer rectangle is a pushout.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $A$ denote the partially ordered set $([2] \times [1] ) \setminus \{ (0,0) \} $. Note that the inclusion maps

\[ A \setminus \{ (2,0), (2,1) \} \hookrightarrow A \quad \quad A \setminus \{ (1,0), (1,1) \} \hookrightarrow A \setminus \{ (1,0) \} \]

admit right adjoints, and therefore induce left cofinal morphisms

\[ \operatorname{N}_{\bullet }( A \setminus \{ (2,0), (2,1) \} ) \hookrightarrow \operatorname{N}_{\bullet }(A) \quad \quad \operatorname{N}_{\bullet }( A \setminus \{ (1,0), (1,1) \} ) \hookrightarrow \operatorname{N}_{\bullet }(A \setminus \{ (1,0) \} ) \]

(Corollary 7.2.3.7). Applying Corollary 7.2.2.3, we obtain the following:

  • The left square of (7.59) is a pullback diagram if and only if $\sigma $ is a limit diagram.

  • The outer rectangle of (7.59) is a pullback diagram if and only if the restriction $\sigma |_{ \operatorname{N}_{\bullet }( ([2] \times [1] ) \setminus \{ (1,0) \} )}$ is a limit diagram.

If the right square of (7.59) is a pullback diagram, then $\sigma |_{\operatorname{N}_{\bullet }(A)}$ is right Kan extended from $\sigma |_{ \operatorname{N}_{\bullet }( A \setminus \{ (1,0) \} )}$, so the desired equivalence follows from Corollary 7.3.7.2. $\square$

Proposition 7.6.3.17 (Rewriting Limits as Pullbacks). Suppose we are given a categorical pushout square of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ K \ar [r] \ar [d] & K_0 \ar [d] \\ K_1 \ar [r] & K_{01}. } \]

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, $K_0$-indexed limits, and $K_1$-indexed limits, then it also admits $K_{01}$-indexed limits. Moreover, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which preserves pullback squares, $K$-indexed limits, $K_0$-indexed limits, and $K_1$-indexed limits, then $F$ also preserves $K$-indexed limits.

Proof. Combine Corollary 7.5.8.5 with (the dual of) Corollary 7.5.8.13. $\square$

Corollary 7.6.3.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits finite limits if and only if it admits pullbacks and has a final object. If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves finite limits if and only if it preserves pullbacks and final objects.

Proof. We will prove the first assertion; the second follows by a similar argument. Assume that the $\infty $-category $\operatorname{\mathcal{C}}$ admits pullbacks and has a final object; we wish to show that $\operatorname{\mathcal{C}}$ admits $K$-indexed limits for every finite simplicial set $K$ (the converse is immediate from the definitions). We proceed by induction on the dimension of $K$. If $K$ is empty, then the desired result follows from our assumption that $\operatorname{\mathcal{C}}$ has an initial object. Let us therefore assume that $K$ has dimension $n \geq 0$, and proceed also by induction on the number of nondegenerate $n$-simplices of $K$. It follows from Proposition 1.1.3.13 that there exists a pushout square of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d] \\ K' \ar [r] & K, } \]

where $K'$ is a simplicial subset of $K$. Since the horizontal maps are monomorphisms, this pushout square is also a categorical pushout square (Example 4.5.4.12). By virtue of Proposition 7.6.3.17, it will suffice to show that the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K'$-indexed limits, $\operatorname{\partial \Delta }^{n}$-indexed limits, and $\Delta ^{n}$-indexed limits. In the first two cases, this follows from our inductive hypothesis. To handle the third case, we observe that the inclusion $\{ 0\} \hookrightarrow \Delta ^ n$ is left cofinal (Example 4.3.7.11). Using Corollary 7.2.2.12, we are reduced to proving that $\operatorname{\mathcal{C}}$ admits $\Delta ^0$-indexed limits, which is immediate (see Example 7.1.1.5). $\square$