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7.4.8 Relative Colimits for Cocartesian Fibrations

Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $D \in \operatorname{\mathcal{D}}$ be an object, and suppose we are given a morphism

\[ f: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}. \]

If $f$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, then it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$. The converse holds if $U$ is a cartesian fibration (Corollary 7.1.7.20), but not in general. In this section, we study the dual situation where $U$ is a cocartesian fibrations. Our main result asserts that $f$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a transport-stable colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$: that is, for every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ carries $f$ to a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D'}$ (Proposition 7.4.8.2). We begin by showing that the collection of $U$-colimit diagrams is stable under covariant transport.

Proposition 7.4.8.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $K$ be a simplicial set, and let $\alpha : F_0 \rightarrow F_1$ be a natural transformation between diagrams $F_0, F_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. Suppose that, for every vertex $x \in K^{\triangleright }$, the morphism $\alpha _ x: F_0(x) \rightarrow F_1(x)$ is $U$-cocartesian. Then:

$(1)$

If $F$ is a $U$-colimit diagram, then $F'$ is also a $U$-colimit diagram.

$(2)$

If $F$ is a $U$-colimit diagram and the natural transformation $\alpha $ carries the cone point $v \in K^{\triangleright }$ to an isomorphism $\alpha _{ v}: F_0( v) \rightarrow F_1( v)$, then $F_0$ is a $U$-colimit diagram.

Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $i: K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty $-category. It follows that the induced map $i^{\triangleright }: K^{\triangleright } \hookrightarrow \operatorname{\mathcal{K}}^{\triangleright }$ is also inner anodyne (Example 4.3.6.7), so that the restriction map $\operatorname{Fun}( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}})$ is a trivial Kan fibration of simplicial sets (Proposition 1.4.7.6). We can therefore lift $\alpha $ to a natural transformation $\overline{\alpha }: \overline{F}_0 \rightarrow \overline{F}_1$ between natural transformations $\overline{F}_0, \overline{F}_1: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Since $i^{\triangleright }$ is bijective on vertices, the natural transformation $\overline{\alpha }$ carries each object of $\operatorname{\mathcal{K}}^{\triangleright }$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$. The morphism $i^{\triangleright }$ is right cofinal (Corollary 7.2.1.11), so Corollary 7.2.2.2 guarantees that $F_0$ is a $U$-colimit diagram if and only if $\overline{F}_0$ is a $U$-colimit diagram. Similarly, $F_1$ is a $U$-colimit diagram if and only if $\overline{F}_1$ is a $U$-colimit diagram. We may therefore replace $\alpha $ by $\overline{\alpha }$ in the statement of Proposition 7.4.8.1, and thereby reduce to the case where $K = \operatorname{\mathcal{K}}$ is an $\infty $-category.

Let us identify $\alpha $ with a functor of $\infty $-categories $F: \Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. For each object $x \in \operatorname{\mathcal{K}}^{\triangleright }$, let us write $x_0 = (0,x)$ and $x_1 = (1,x)$ for the corresponding objects of $\Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright }$, so that the inclusion map $\Delta ^1 \times \{ x\} \hookrightarrow \Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright }$ determines a morphism $e_{x}: x_0 \rightarrow x_1$. By construction, the functor $F$ carries each $e_{x}$ to the $U$-cocartesian morphism $\alpha _ x$ of $\operatorname{\mathcal{C}}$. By virtue of Proposition 7.1.3.8, $e_{x}$ is final when viewed as an object of the $\infty $-category

\[ ( \{ 0\} \times \operatorname{\mathcal{K}}^{\triangleright } ) \times _{ ( \Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright } ) } (\Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright } )_{/x_1} \simeq (\operatorname{\mathcal{K}}^{\triangleright })_{/x}, \]

so that $F$ is $U$-left Kan extended from $\{ 0\} \times \operatorname{\mathcal{K}}^{\triangleright }$ at $x_1$ (Corollary 7.2.3.6). Allowing the object $x$ to vary, we see that the functor $F$ is $U$-left Kan extended from $\{ 0\} \times \operatorname{\mathcal{K}}^{\triangleright }$.

We now prove $(1)$. Suppose that $F_0$ is a $U$-colimit diagram. Then it is $U$-left Kan extended from $\operatorname{\mathcal{K}}$ (Example 7.4.3.7). Applying Proposition 7.4.7.6, we see that the functor $F$ is $U$-left Kan extended from $\{ 0\} \times \operatorname{\mathcal{K}}$, and therefore from the larger subcategory $\Delta ^1 \times \operatorname{\mathcal{K}}\subseteq \Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright }$. It follows that the composite map

\[ ( \Delta ^1 \times \operatorname{\mathcal{K}}) \star \{ v_1\} \hookrightarrow \Delta ^1 \times (\operatorname{\mathcal{K}}\star \{ v\} ) \xrightarrow {F} \operatorname{\mathcal{C}} \]

is a $U$-colimit diagram. Since the inclusion map $\{ 1\} \times \operatorname{\mathcal{K}}\hookrightarrow \Delta ^1 \times \operatorname{\mathcal{K}}$ is right cofinal (Proposition 7.2.1.3), Corollary 7.2.2.2 guarantees that $F_1 = F|_{ (\{ 1\} \times \operatorname{\mathcal{K}}) \star \{ v_1\} }$ is also a $U$-colimit diagram.

We now prove $(2)$. Let $\pi : \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \Delta ^1$ be the functor carrying $\operatorname{\mathcal{K}}$ to the vertex $0 \in \Delta ^1$ and the cone point $v \in \operatorname{\mathcal{K}}^{\triangleright }$ to the vertex $1 \in \Delta ^1$, and let $G: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the functor given by the composition

\[ \operatorname{\mathcal{K}}^{\triangleright } \xrightarrow { (\pi , \operatorname{id}) } \Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \Delta ^1. \]

Note that there is a natural transformation $\beta : F_0 \rightarrow G$ which is the identity when restricted to $\operatorname{\mathcal{K}}$ and which carries the cone point $v$ to the morphism $\alpha _{v}: F_0(v) \rightarrow F_1(v) = G(v)$. If $\alpha _{v}$ is an isomorphism, then the natural transformation $\beta $ is also an isomorphism (Theorem 4.4.4.4). Consequently, to show that $F_0$ is a $U$-colimit diagram, it will suffice to show that $G$ is a $U$-colimit diagram (Proposition 7.1.7.13). Arguing as above, we see that the functor $F|_{ \Delta ^1 \times \operatorname{\mathcal{K}}}$ is $U$-left Kan extended from the full subcategory $\{ 0\} \times \operatorname{\mathcal{K}}\subseteq \Delta ^1 \times \operatorname{\mathcal{K}}$. Applying Proposition 7.4.7.1, we see that $G$ is a $U$-colimit diagram if and only if the composite map

\[ ( \Delta ^1 \times \operatorname{\mathcal{K}}) \star \{ v_1\} \hookrightarrow \Delta ^1 \times (\operatorname{\mathcal{K}}\star \{ v\} ) \xrightarrow {F} \operatorname{\mathcal{C}} \]

is a $U$-colimit diagram. By virtue of Corollary 7.2.2.2, this is equivalent to the requirement that $F_1$ is a $U$-colimit diagram. $\square$

Proposition 7.4.8.2. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories, let $D \in \operatorname{\mathcal{D}}$ be an object, and let $f: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ be a diagram. Then $f$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$ if and only if it satisfies the following condition:

$(\ast )$

Let $e: D \rightarrow D'$ be a morphism in the $\infty $-category $\operatorname{\mathcal{D}}$ and let $e_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ be the covariant transport functor of Notation 5.2.2.5. Then $(e_{!} \circ f): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D'}$ is a colimit diagram in the $\infty $-category

Example 7.4.8.3. In the situation of Proposition 7.4.8.2, suppose that that the cocartesian fibration $U$ is also a cartesian fibration. Then, for every morphism $e: D \rightarrow D'$ of $\operatorname{\mathcal{D}}$, the covariant transport functor $e_{!}$ has a right adjoint $e^{\ast }$, given by contravariant transport along $e$ (Proposition 6.2.3.4). In particular, the functor $e_{!}$ automatically preserves $K$-indexed colimits (Corollary 7.1.5.20). We therefore recover the criterion of Corollary 7.1.7.20: the morphism $f$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$.

Proof of Proposition 7.4.8.2. For every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, we can choose a natural transformation $\alpha : f \rightarrow e_{!} \circ f$ carrying each vertex of $K^{\triangleright }$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$. It follows from Proposition 7.4.8.1 that if $f$ is a $U$-colimit diagram, then $e_{!} \circ f$ is also a $U$-colimit diagram, and therefore a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D'}$ (Corollary 7.1.7.20). This proves the necessity of condition $(\ast )$. For the converse, suppose that $f$ satisfies condition $(\ast )$; we wish to show that $f$ is a $U$-colimit diagram. By virtue of Proposition 7.1.7.12, this is equivalent to the assertion that for every object $C \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

7.55
\begin{equation} \begin{gathered}\label{equation:relative-colimit-by-fiber} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( f, \underline{C}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f|_{K}, \underline{C}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f|_{K}, U \circ \underline{C}|_{K}) } \end{gathered} \end{equation}

is a homotopy pullback square, where $\underline{C} \in \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$ is the constant diagram taking the value $C$. Since $U$ is an inner fibration, the vertical maps in this diagram are Kan fibrations (Proposition 4.6.1.19). Using the criterion of Example 3.4.1.4, it will suffice to show that for every vertex $u \in \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C} )$, the induced map

\[ \xymatrix@R =50pt@C=50pt{ \{ u\} \times _{\operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( f, \underline{C})} \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C} ) \ar [d]^{\theta _ u} \\ \{ u\} \times _{ \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f|_{K}, U \circ \underline{C}|_{K}) } \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f|_{K}, \underline{C}|_{K} )} \]

is a homotopy equivalence of Kan complexes. Set $D' = U(C)$, so that $u$ can be identified with a morphism of simplicial sets $K^{\triangleright } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' )$, and that the condition that $\theta _ u$ is a homotopy equivalence depends only on the homotopy class of $u$. Since the simplicial set $K^{\triangleright }$ is weakly contractible (Example 4.3.7.11, we may assume without loss of generality that $u: K^{\triangleright } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' )$ is the constant map taking the value $e$, for some morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$. In this case, we can use Proposition 5.1.2.1 to identify $\theta _{u}$ with the restriction map

\[ \operatorname{Hom}_{ \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}_{D'})}( e_{!} \circ f, \underline{D} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( K, \operatorname{\mathcal{C}}_{D'})}( e_{!} \circ f_{K}, \underline{D}|_{K} ), \]

which is a homotopy equivalence by virtue of assumption $(\ast )$ (see Proposition 7.1.7.12). $\square$

Using Proposition 7.4.8.2, we obtain a relative version of Corollary 7.2.3.8:

Corollary 7.4.8.4. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories, let $K$ be a weakly contractible simplicial set, and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a diagram. The following conditions are equivalent:

$(1)$

The diagram $\overline{f}$ carries each edge of $K^{\triangleright }$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$.

$(2)$

The restriction $f = \overline{f}|_{K}$ carries each edge of $K$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$, and $\overline{f}$ is a $U$-colimit diagram.

Proof. Without loss of generality, we may assume that $f$ carries each edge of $K$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$. Let $\pi : \Delta ^1 \times K^{\triangleright } \rightarrow K^{\triangleright }$ be the morphism which is the identity on $\{ 0\} \times K^{\triangleright }$ and which carries $\{ 1\} \times K^{\triangleright }$ to the cone point $v \in K^{\triangleright }$. Set $C = f(v) \in \operatorname{\mathcal{C}}$ and $D = U(C) \in \operatorname{\mathcal{D}}$. Proposition 5.2.1.3 guarantees that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times K^{\triangleright } \ar [r]^-{ \overline{f}} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \Delta ^1 \times K^{\triangleright } \ar [r]^-{U \circ \overline{f} \circ \pi } \ar@ {-->}[ur]^{ \alpha } & \operatorname{\mathcal{D}}} \]

admits a solution $\alpha : \Delta ^1 \times K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which carries $\Delta ^1 \times \{ x\} $ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$, for each vertex $x \in K^{\triangleright }$. Set $\overline{g} = \alpha |_{ \{ 1\} \times K^{\triangleright }}$, which we regard as a morphism from $K^{\triangleright }$ to the $\infty $-category $\operatorname{\mathcal{C}}_{D}$, and let us identify $\alpha $ with a natural transformation from $\overline{f}$ to $\overline{g}$. Note that $\alpha _{v}: \overline{f}(v) \rightarrow \overline{g}(v)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$ satisfying $U(\alpha _ v) = \operatorname{id}_{D}$, and is therefore an isomorphism (Proposition 5.1.1.8). Applying Proposition 7.4.8.1, we can reformulate $(2)$ as follows:

$(2')$

The morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$.

Set $g = \overline{g}|_{K}$. For every edge $u: x \rightarrow y$ of $K$, we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ f(x) \ar [r]^-{ f(u) } \ar [d]^{ \alpha _ x} & f(y) \ar [d]^{ \alpha _ y} \\ g(x) \ar [r]^-{ g(u) } & g(y) } \]

where $f(u)$, $\alpha _ x$, and $\alpha _ y$ are $U$-cocartesian. Applying Corollary 5.1.2.4, we deduce that $g(u)$ is $U$-cocartesian when viewed as a morphism of $\operatorname{\mathcal{C}}$, and is therefore an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$ (Proposition 5.1.1.8). Similarly, for every vertex $x \in K$, the unique edge $c_{x}: x \rightarrow v$ of $K^{\triangleright }$ determines a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ f(x) \ar [r]^-{\overline{f}(c_ x)} \ar [d]^{ \alpha _ x} & \overline{f}(v) \ar [d]^{ \alpha _ v} \\ g(x) \ar [r]^-{ \overline{g}(c_ x)} & \overline{g}(v), } \]

where $\alpha _{x}$ is $U$-cocartesian and $\alpha _{v}$ is an isomorphism. Combining Corollary 5.1.2.4, Corollary 5.1.2.5, and Proposition 5.1.1.8, we see that $\overline{f}(c_ x)$ is $U$-cocartesian if and only if $\overline{g}(c_ x)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$. We can therefore reformulate condition $(1)$ as follows:

$(1')$

The diagram $\overline{g}$ carries each edge of $K^{\triangleright }$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$.

By virtue of Corollary 7.2.3.8, $(1')$ is equivalent to the requirement that $\overline{g}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$. In particular, the implication $(1') \Rightarrow (2')$ follows from Corollary 7.1.7.20. To prove the converse, it will suffice to show that condition $(1')$ is satisfied, then for every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ carries $\overline{g}$ to a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D'}$ (Proposition 7.4.8.2). This follows immediately from Corollary 7.2.3.8 (applied to the composite diagram $K^{\triangleleft } \xrightarrow { \overline{g} } \operatorname{\mathcal{C}}_{D} \xrightarrow { e_{!} } \operatorname{\mathcal{C}}_{D'}$). $\square$

The criterion of Proposition 7.4.8.2 has a counterpart for the existence of $U$-colimit diagrams.

Proposition 7.4.8.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories, and suppose we are given a lifting problem

7.56
\begin{equation} \begin{gathered}\label{equation:relative-colimit-existence-criterion} \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{f_0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleright } \ar [r]^-{g } \ar@ {-->}[ur]^{ \overline{f}_0 } & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

Let $v \in K^{\triangleright }$ be the cone point and set $D = g(v)$. Then there exists a diagram $f_1: K \rightarrow \operatorname{\mathcal{C}}_{D} \subseteq \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : f_0 \rightarrow f_1$ which carries each vertex $x \in K$ to a $U$-cocartesian morphism $\alpha _ x: f_0(x) \rightarrow f_1(x)$ of $\operatorname{\mathcal{C}}$, where$U \circ \alpha $ is given by the composition $\Delta ^1 \times K \xrightarrow {c} K^{\triangleright } \xrightarrow {g} \operatorname{\mathcal{D}}$. Moreover, the lifting problem (7.56) admits a solution $\overline{f}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram if and only if the following pair of conditions is satisfied:

$(1)$

The diagram $f_1$ admits a colimit $\overline{f}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$.

$(2)$

Let $e: D \rightarrow D'$ be a morphism in the $\infty $-category $\operatorname{\mathcal{D}}$ and let $e_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ be the covariant transport functor of Notation 5.2.2.5. Then $(e_{!} \circ \overline{f}_1): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D'}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D'}$.

Proof. The existence (and essential uniqueness) of the diagram $f_1$ and the natural transformation $\alpha : f_0 \rightarrow f_1$ follow from Proposition 5.2.1.3. Let us first show that conditions $(1)$ and $(2)$ are necessary. Suppose that the lifting problem (7.56) admits a solution $\overline{f}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram. Using Proposition 5.2.1.3, we can extend $f_1$ to a diagram $\overline{f}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ and $\alpha $ to a natural transformation $\overline{\alpha }: \overline{f}_0 \rightarrow \overline{f}_1$ which carries each vertex $x \in K^{\triangleright }$ to a $U$-cocartesian morphism $\overline{\alpha }_ x: \overline{f}_0(x) \rightarrow \overline{f}_1(x)$. Proposition 7.4.8.1 guarantees that $\overline{f}_{1}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, and therefore satisfies conditions $(1)$ and $(2)$ by virtue of Proposition 7.4.8.2.

Now suppose that conditions $(1)$ and $(2)$ are satisfied. Let $\overline{f}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ be a colimit diagram extending $f_1$. It follows from $(2)$ that $\overline{f}_1$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Let $\pi : ( \Delta ^1 \times K )^{\triangleright } \rightarrow K^{\triangleright }$ denote the morphism which is the identity when restricted to $\{ 0\} \times K$, and which carries $(\{ 1\} \times K)^{\triangleright }$ to the cone point of $K^{\triangleright }$. Since the inclusion map $\{ 1 \} \times K \hookrightarrow \Delta ^1 \times K$ is right cofinal (Proposition 7.2.1.3), Proposition 7.2.2.7 guarantees that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times K \ar [r]^-{\alpha } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ (\Delta ^1 \times K)^{\triangleright } \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{\overline{\alpha }} & \operatorname{\mathcal{D}}} \]

admits a solution $\overline{\alpha }: (\Delta ^1 \times K)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram. Note that in this case $\overline{f}'_1 = \overline{\alpha }|_{ (\{ 1\} \times K)^{\triangleright } }$ is also a $U$-colimit diagram (Corollary 7.2.2.2). Setting $\overline{f}_0 = \overline{\alpha }|_{ (\{ 0\} \times K)^{\triangleright } }$, we note that $\overline{\alpha }$ determines a natural transformation of functors $\overline{f}_0 \rightarrow \overline{f}'_1$ which carries each vertex of $x$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$ and carries the cone point to an identity morphism of $\operatorname{\mathcal{C}}$. Applying the criterion of Proposition 7.4.8.1, we conclude that $\overline{f}_0$ is a $U$-colimit diagram which solves the lifting problem (7.56). $\square$

Corollary 7.4.8.6. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories and let $K$ be a simplicial set. The following conditions are equivalent:

$(1)$

For every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ admits $K$-indexed colimits. Moreover, for every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ preserves $K$-indexed colimits.

$(2)$

Every lifting problem

7.57
\begin{equation} \begin{gathered}\label{equation:all-relative-colimits} \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{f} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

admits a solution $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram.

Proof. The implication $(1) \Rightarrow (2)$ follows immediately from Proposition 7.4.8.5. Conversely, suppose that $(2)$ is satisfied. For each object $D \in \operatorname{\mathcal{D}}$, condition $(2)$ guarantees that every diagram $f: K \rightarrow \operatorname{\mathcal{C}}_{D}$ admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ which is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$. In particular, $\overline{f}$ is a colimit diagram in $\operatorname{\mathcal{C}}_{D}$ (Corollary 7.1.7.20) having the property that for every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the composition $e_{!} \circ \overline{f}$ is a colimit diagram in $\operatorname{\mathcal{C}}_{D'}$ (Proposition 7.4.8.2). To complete the proof, we observe that if $\overline{f}': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ is any other colimit diagram satisfying $\overline{f}'|_{K} = f$, then $\overline{f}'$ is isomorphic to $\overline{f}$ as an object of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}_{D} )$, so that $e_{!} \circ \overline{f}'$ is also a colimit diagram in $\operatorname{\mathcal{C}}_{D'}$ (Corollary 7.1.5.7). $\square$

Corollary 7.4.8.7. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a cocartesian fibration of $\infty $-categories, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that the following conditions are satisfied:

  • For every object $C \in \operatorname{\mathcal{C}}$ and every object $E \in \operatorname{\mathcal{E}}$, the $\infty $-category $\operatorname{\mathcal{D}}_{E} = \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ admits $\operatorname{\mathcal{C}}^{0}_{/C}$-indexed colimits.

  • For every object $C \in \operatorname{\mathcal{C}}$ and every morphism $e: E \rightarrow E'$ in $\operatorname{\mathcal{E}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{D}}_{E} \rightarrow \operatorname{\mathcal{D}}_{E'}$ preserves $\operatorname{\mathcal{C}}^{0}_{/C}$-indexed colimits.

Then every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [r]^-{ F } \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur]^{\overline{F}} & \operatorname{\mathcal{E}}} \]

admits a solution $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. Combine Proposition 7.4.5.3 with Corollary 7.4.8.6. $\square$