Definition 7.6.2.1. 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 7.1.3.4), and that $\sigma $ is a pushout square if it is a colimit diagram in $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
7.6.2 Pullback and Pushout Squares
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 1.5.2.15). 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) \} $).
Example 7.6.2.2. 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 is a pullback square in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 7.6.2.1); this follows from Example 7.1.1.4 and Remark 7.1.3.6. 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 $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
\[ \operatorname{N}_{\bullet }(\sigma ): \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \]
Warning 7.6.2.3. 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 Beware that, if $\sigma $ is a pullback square in the $\infty $-category $\operatorname{\mathcal{C}}$, then the associated diagram need not be a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (see Example 7.6.2.4 and Exercise 7.6.2.5). If $Y$ is an object of $\operatorname{\mathcal{C}}$, then the map of sets is surjective, but need not be injective. Given a commutative diagram 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 (7.61) to a commutative diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Such a lift always exists (Exercise 1.5.2.10), but is not unique (even up to homotopy).
\[ \xymatrix@C =50pt@R=50pt{ X_{01} \ar [r]^-{g_0} \ar [d]^-{g_1} & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]
\[ \xymatrix@C =50pt@R=50pt{ X_{01} \ar [r]^-{[g_0]} \ar [d]^-{[g_1]} & X_0 \ar [d]^-{[f_0]} \\ X_1 \ar [r]^-{[f_1]} & X } \]
\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X_{01} ) \xrightarrow { ( [g_0], [g_1]) \circ } \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X_0) \times _{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X ) } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y, X_1 ) \]
7.61
\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}
Example 7.6.2.4. 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 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 7.6.3.2). However, if $X$ is contractible and $X_{s}$ is not, then (7.62) is not a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.
7.62
\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}
Exercise 7.6.2.5. 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
Show that (7.63) 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 7.6.3.2).
Show that, if $H$ is contained in the center of $G$, then the diagram (7.63) 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 (7.63) is not a pullback square in $\mathrm{h} \mathit{\operatorname{Kan}}$.
7.63
\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}
Variant 7.6.2.6. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that a diagram $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ is a $U$-pullback square if it is a $U$-limit diagram in $\operatorname{\mathcal{C}}$ (Definition 7.1.6.1). We say that $\sigma $ is a $U$-pushout square if it is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.
Remark 7.6.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square in $\operatorname{\mathcal{C}}$. Then $\sigma $ is a pullback square if and only if it is $U$-pullback square, where $U: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is the projection map (see Example 7.1.6.3).
Remark 7.6.2.8 (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. More generally, if $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories, then $\sigma $ is a $U$-pullback square if and only if $\sigma '$ is a $U$-pullback square, and $\sigma $ is a $U$-pushout square if and only if $\sigma '$ is a $U$-pushout square.
Remark 7.6.2.9. 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 7.6.2.8). More generally, if $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories, then $\sigma $ is a $U$-pushout diagram if and only if $\sigma ^{\operatorname{op}}$ is a $U^{\operatorname{op}}$-pullback diagram.
Remark 7.6.2.10. 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. More generally, if $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories, then then $\sigma $ is a $U$-pullback square if and only if $\sigma '$ is a $U$-pullback square, and $\sigma $ is a $U$-pushout square if and only if $\sigma '$ is a $U$-pushout square (see Proposition 7.1.6.15).
Notation 7.6.2.11 (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 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 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 $Y$.
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X } \]
\[ \xymatrix@R =50pt@C=50pt{ Y \ar [r]^-{g_0} \ar [d]^-{g_1} & Y_0 \ar [d] \\ Y_1 \ar [r] & Y_{01} } \]
Definition 7.6.2.12. 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 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 We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves pushouts 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 $\operatorname{\mathcal{D}}$.
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]
\[ \xymatrix@R =50pt@C=50pt{ Y \ar [r]^-{g_0} \ar [d]^-{g_1} & Y_0 \ar [d] \\ Y_1 \ar [r] & Y_{01}. } \]
Remark 7.6.2.13. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration of $\infty $-categories, and suppose that $\operatorname{\mathcal{D}}$ admits pullbacks. Then $\operatorname{\mathcal{C}}$ also admits pullbacks, and the functor $U$ preserves pullbacks. See Corollary 7.1.6.20.
Proposition 7.6.2.14. 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 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 $\operatorname{\mathcal{C}}_{/X}$.
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X. } \]
Proof. This is a special case of Remark 7.1.3.11. $\square$
Notation 7.6.2.15 (Relative Diagonals). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: Y \rightarrow X$ be a morphism in $\operatorname{\mathcal{C}}$. Suppose that there exists a pullback square in the $\infty $-category $\operatorname{\mathcal{C}}$. Let us abuse notation by identifying $Y$ with an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$, so that $Y \times _{X} Y$ can be viewed as a product of $Y$ with itself in $\operatorname{\mathcal{C}}_{/X}$ (Proposition 7.6.2.14). Applying the construction of Notation 7.6.1.13, we obtain a morphism $\delta _{Y/X}: Y \rightarrow Y \times _{X} Y$, which we will refer to as the relative diagonal of $f$. It is characterized (up to homotopy) by the requirement that ( 7.64) can be extended to a commutative diagram where the outer square is the commutative diagram given by the composition
7.64
\begin{equation} \begin{gathered}\label{equation:relative-diagonal} \xymatrix { Y \times _{X} Y \ar [r]^{ \pi } \ar [d]^{\pi '} & Y \ar [d]^{f} \\ Y \ar [r]^{f} & X } \end{gathered} \end{equation}
\[ \xymatrix { Y \ar [dr]^{ \delta _{Y/X} } \ar [drr]^{\operatorname{id}_ Y} \ar [ddr]^{ \operatorname{id}_{Y} } & & \\ & Y \times _{X} Y \ar [r]^{\pi } \ar [d]^{\pi '} & Y \ar [d]^{f} \\ & Y \ar [r]^{f} & X, } \]
\[ \Delta ^1 \times \Delta ^1 \xrightarrow {(i,j) \mapsto ij} \Delta ^1 \xrightarrow {f} \operatorname{\mathcal{C}}. \]
Variant 7.6.2.16 (Relative Codiagonals). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: Y \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$, and suppose that there exists a pushout square Applying the construction of Notation 7.6.2.15 in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, we obtain a morphism $\gamma _{Y/X}: X \coprod _{Y} X \rightarrow X$ which we will refer to as the relative codiagonal of the morphism $f$.
\[ \xymatrix { Y \ar [r]^{f} \ar [d]^{f} & X \ar [d] \\ X \ar [r] & X \coprod _{Y} X. } \]
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 7.6.2.17. 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 7.6.2.14, 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 4.6.7.22), this is equivalent to the requirement that $\operatorname{\mathcal{C}}_{/X}$ admits finite products (Corollary 7.6.1.23). $\square$
Remark 7.6.2.18. 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 7.6.2.19. 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 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 product of $X_0$ with $X_1$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f_0} \ar [d]^-{f_1} & X_0 \ar [d] \\ X_1 \ar [r] & \mathbf{1}. } \]
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 4.6.7.10), so that the desired result follows from the criterion of Proposition 7.6.2.14. $\square$
It will sometimes be convenient to work with an alternative characterization of fiber products. Recall that, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the evaluation functor
\[ \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad (f: X \rightarrow Y) \mapsto Y \]
is a cocartesian fibration (Example 5.3.7.5), whose fiber over each object $X \in \operatorname{\mathcal{C}}$ is the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} $ (which is equivalent to the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$).
Proposition 7.6.2.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the functor given by evaluation at the vertex $1 \in \Delta ^1$, and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a diagram, which we display informally as be a diagram in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The diagram $\sigma $ is a pullback square (Definition 7.6.2.1).
Let $u: f' \rightarrow f$ be the morphism in $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ determined by $\sigma $. Then $u$ is $\operatorname{ev}_{1}$-cartesian.
The morphism $u: f' \rightarrow f$ is locally $\operatorname{ev}_{1}$-cartesian.
\[ \xymatrix { X' \ar [r]^{g'} \ar [d]^{f'} & X \ar [d]^{f} \\ Y' \ar [r]^{g} & Y } \]
Remark 7.6.2.21. In the formulation of Proposition 7.6.2.20, the diagram $\sigma $ can also be viewed as a morphism $v: g' \rightarrow g$ in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (by interchanging the factors of $\Delta ^1 \times \Delta ^1$). Combining Proposition 7.6.2.20 with Remark 7.6.2.8, we see that $u$ is $\operatorname{ev}_{1}$-cartesian if and only if $v$ is $\operatorname{ev}_{1}$-cartesian.
Proof of Proposition 7.6.2.20. Since the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration (Example 5.3.7.5), the equivalence $(2) \Leftrightarrow (3)$ is a special case of Lemma 6.2.3.5. We will prove that $(1) \Leftrightarrow (2)$. Let $K$ denote the nerve of the partially ordered set $\{ (0,1) < (1,1) > (1,0) \} $, which we regard as a full subcategory of $\Delta ^1 \times \Delta ^1$, and let $U: \operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ be the restriction map. Then $\sigma $ is a pullback diagram in $\operatorname{\mathcal{C}}$ if and only if it is $U$-final when viewed as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}})$ (Proposition 7.1.7.4). Unwinding the definitions, we have a pullback diagram of simplicial sets
\[ \xymatrix { \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \{ f \} \ar [r] \ar [d]^{U_0} & \operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}}) \ar [d]^{U} \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}} \{ Y\} & \operatorname{Fun}(K, \operatorname{\mathcal{C}}), } \]
and the morphism $u$ is $\operatorname{ev}_{1}$-cartesian if and only if it is $U_0$-final when viewed as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \{ f \} $ (Example 7.1.6.9). The equivalence $(1) \Leftrightarrow (2)$ now follows from the criterion of Corollary 7.1.5.21, since $U$ is a cocartesian fibration (Corollary 5.3.7.4). $\square$
Definition 7.6.2.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that a functor $f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X}$ is given by pullback along $f$ if it is a right adjoint to the functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ given by postcomposition with $f$ (see Example 4.3.6.15). Note that this condition characterizes the functor $f^{\ast }$ up to isomorphism (see Remark 6.2.1.19).
Variant 7.6.2.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. By virtue of Example 5.3.7.5, the evaluation functor $\operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of $\infty $-categories, and therefore determines a covariant transport functor We say that a functor is given by pullback along $f$ if it is right adjoint to the functor $f_{!}$. By virtue of Proposition 5.3.7.6, this is equivalent to the existence of a homotopy commutative diagram where the vertical maps are the pinch inclusion equivalences of Corollary 4.6.4.18 and the upper horizontal map is given by pullback along $f$ (in the sense of Definition 7.6.2.22).
\[ f_{!}: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}} \{ Y\} . \]
\[ f^{\ast }: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \]
\[ \xymatrix { \operatorname{\mathcal{C}}_{/Y} \ar@ {-->}[r] \ar [d]^{ \delta _{/Y} }_{\sim } & \operatorname{\mathcal{C}}_{ / X} \ar [d]^{ \delta _{/X} }_{\sim } \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \ar [r]^{ f^{\ast } } & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} , } \]
Proposition 7.6.2.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
There exists a functor $f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X}$ given by pullback along $f$ (in the sense of Definition 7.6.2.22).
There exists a functor $f^{\ast }: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} $ given by pullback along $f$ (in the sense of Variant 7.6.2.23).
The left vertical map of the pullback square
is a cartesian fibration of $\infty $-categories.
For every morphism $u: Y' \rightarrow Y$, there exists a pullback diagram
in the $\infty $-category $\operatorname{\mathcal{C}}$.
\[ \xymatrix { \Delta ^1 \times _{\operatorname{\mathcal{C}}} \Delta ^1 \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{ev}_1 } \\ \Delta ^{1} \ar [r]^{f} & \operatorname{\mathcal{C}}} \]
\[ \xymatrix { X' \ar [r] \ar [d] & Y' \ar [d]^{u} \\ X \ar [r]^{f} & Y } \]
Moreover, if these conditions are satisfied, then both pullback functors are given on objects by the construction $Y' \mapsto X \times _{Y} Y'$.
Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 5.3.7.6, the equivalence $(2) \Leftrightarrow (3)$ from Proposition 6.2.3.4, and the equivalence $(3) \Leftrightarrow (4)$ from Propositions 7.6.2.20. $\square$
Corollary 7.6.2.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
For every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the composition functor
of Example 4.3.6.15 admits a right adjoint.
For every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the composition functor
admits a right adjoint.
The evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration of $\infty $-categories.
The evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibraton of $\infty $-categories.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits pullbacks.
\[ \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y} \quad \quad e \mapsto (f \circ e) \]
\[ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \quad \quad e \mapsto (f \circ e) \]
Proof. The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3) \Leftrightarrow (4)$ follow from Proposition 7.6.2.24 by allowing the morphism $f$ to vary. The equivalence of $(3)$ and $(3')$ follows from Proposition 7.6.2.20. $\square$
Proposition 7.6.2.26. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square, represented informally by the diagram Then:
If $f$ is $U$-cartesian, then $\sigma $ is a $U$-pullback square if and only if $f'$ is also $U$-cartesian.
If $f'$ is $U$-cocartesian, then $\sigma $ is a $U$-pushout square if and only if $f$ is also $U$-cocartesian.
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r]^-{f'} \ar [d] & Y' \ar [d] \\ X \ar [r]^-{f} & Y. } \]
Proof. We will prove $(1)$; the proof of $(2)$ is similar. Note that $\sigma $ restricts to a diagram
\[ \sigma _0: \operatorname{N}_{\bullet }( \{ (0,1) < (1,1) > (1,0) \} ) \rightarrow \operatorname{\mathcal{C}} \]
satisfying $\sigma _0( 0,1) = X$, $\sigma _0( 1,1) = Y$, and $\sigma _0( 1,0 ) = Y'$. The assumption that $f$ is $U$-cartesian guarantees that $\sigma _0$ is $U$-right Kan extended from the full subcategory
\[ \{ 1\} \times \Delta ^1 \subseteq \operatorname{N}_{\bullet }( \{ (0,1) < (1,1) > (1,0) \} ). \]
It follows that $\sigma $ is a $U$-pullback diagram if and only if the restriction $\sigma |_{ \operatorname{N}_{\bullet }( \{ (0,0) < (1,0) < (1,1) \} )}$ is a $U$-limit diagram (Proposition 7.3.8.1) By virtue of Corollary 7.2.2.5, this is equivalent to the requirement that $f'$ is $U$-cartesian. $\square$
Corollary 7.6.2.27. 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 Then:
If $f$ is an isomorphism, then $\sigma $ is a pullback square if and only if $f'$ is also an isomorphism.
If $f'$ is an isomorphism, then $\sigma $ is a pushout square if and only if $f$ is also an isomorphism.
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r]^-{f'} \ar [d] & Y' \ar [d] \\ X \ar [r]^-{f} & Y. } \]
Proof. Combine Proposition 7.6.2.26 with Remark 7.6.2.7 (and Example 5.1.1.4). $\square$
Proposition 7.6.2.28 (Transitivity). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\sigma : \Delta ^2 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be diagram, which we depict informally as Then:
Assume that the right square of (7.65) is a $U$-pullback. Then the left square is a $U$-pullback if and only if the outer rectangle is a $U$-pullback.
Assume that the left square of (7.65) is a $U$-pushout. Then the left square is a $U$-pushout if and only if the outer rectangle is a $U$-pushout.
7.65
\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}
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 7.2.2.2, we obtain the following:
The left square of (7.65) is a $U$-pullback diagram if and only if $\sigma $ is a $U$-limit diagram.
The outer rectangle of (7.65) is a $U$-pullback diagram if and only if the restriction $\sigma |_{ \operatorname{N}_{\bullet }( ([2] \times [1] ) \setminus \{ (1,0) \} )}$ is a $U$-limit diagram.
If the right square of (7.65) is a $U$-pullback diagram, then $\sigma |_{\operatorname{N}_{\bullet }(A)}$ is $U$-right Kan extended from $\sigma |_{ \operatorname{N}_{\bullet }( A \setminus \{ (1,0) \} )}$, so the desired equivalence follows from Proposition 7.3.8.1. $\square$
Proposition 7.6.2.29 (Rewriting Limits as Pullbacks). Suppose we are given a categorical pushout square of simplicial sets 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.
\[ \xymatrix@R =50pt@C=50pt{ K \ar [r] \ar [d] & K_0 \ar [d] \\ K_1 \ar [r] & K_{01}. } \]
Proof. Combine Corollary 7.5.8.5 with (the dual of) Corollary 7.5.8.13. $\square$
Corollary 7.6.2.30. 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 a final 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 1.1.4.12 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 7.6.2.29, 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). $\square$
Example 7.6.2.31. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then, for every object $X \in \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ admits finite limits. This follows from Corollary 7.6.2.30, since $\operatorname{\mathcal{C}}_{/X}$ also admits finite pullbacks (Remark 7.6.2.13), and has a final object given by the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ (Proposition 4.6.7.22). Similarly, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor which preserves pullbacks, then the induced functor $F_{/X}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{D}}_{ /F(X)}$ preserves finite limits.