# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

### 7.3.9 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 7.1.5.20), 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 7.3.9.2). We begin by showing that the collection of $U$-colimit diagrams is stable under covariant transport.

Proposition 7.3.9.1. 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 $F_0, F_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. 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 $F_0$ is a $U$-colimit diagram, then $F_1$ is also a $U$-colimit diagram.

$(2)$

If $F_1$ 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 1.5.7.6). 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 7.2.1.13), 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 7.3.9.1, 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 $x \in \operatorname{\mathcal{K}}^{\triangleright }$, we can regard $\Delta ^1 \times \{ x\}$ with a morphism $e_{x}: (0,x) \rightarrow (1,x)$ in the $\infty$-category $\Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright }$. By construction, the functor $F$ carries each $e_{x}$ to the $U$-cocartesian morphism $\alpha _ x$ of $\operatorname{\mathcal{C}}$. By virtue of Proposition 4.6.7.22, $e_{x}$ is final when viewed as an object of the $\infty$-category

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

so that $F$ is $U$-left Kan extended from $\{ 0\} \times \operatorname{\mathcal{K}}^{\triangleright }$ at $(1,x)$ (Corollary 7.2.2.5). 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 $F_0$ is $U$-left Kan extended from $\operatorname{\mathcal{K}}$ (Example 7.3.3.9). Applying Proposition 7.3.8.6, 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

$( \Delta ^1 \times \operatorname{\mathcal{K}}) \star \{ (1,v) \} \hookrightarrow \Delta ^1 \times (\operatorname{\mathcal{K}}\star \{ v\} ) \xrightarrow {F} \operatorname{\mathcal{C}}$

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 $F_1 = F|_{ \{ 1\} \times \operatorname{\mathcal{K}}^{\triangleright } }$ 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

$\operatorname{\mathcal{K}}^{\triangleright } \xrightarrow { (\pi , \operatorname{id}) } \Delta ^1 \times \operatorname{\mathcal{K}}^{\triangleright } \xrightarrow {F} \operatorname{\mathcal{C}}.$

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 7.1.5.13). 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 7.3.8.1, we see that $G$ is a $U$-colimit diagram if and only if the composite map

$( \Delta ^1 \times \operatorname{\mathcal{K}}) \star \{ (1,v) \} \hookrightarrow \Delta ^1 \times (\operatorname{\mathcal{K}}\star \{ v\} ) \xrightarrow {F} \operatorname{\mathcal{C}}$

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 7.3.9.2. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a locally 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 5.2.2.9. Then $(e_{!} \circ f): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D'}$ is a colimit diagram in the $\infty$-category

Example 7.3.9.3. In the situation of Proposition 7.3.9.2, suppose that that $U$ 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 7.1.3.21). We therefore recover the criterion of Corollary 7.1.5.20: 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 7.3.9.2. 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 7.3.9.1 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 7.1.5.20). 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 7.1.5.12, this is equivalent to the assertion that for every object $C \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

7.44
$$\begin{gathered}\label{equation:relative-colimit-by-fiber} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( f, \underline{C}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f|_{K}, \underline{C}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f|_{K}, U \circ \underline{C}|_{K}) } \end{gathered}$$

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 4.6.1.21 and Corollary 4.1.4.3). 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

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

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 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 5.1.3.11 to replace $\theta _{u}$ by 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 7.1.5.12). $\square$

Using Proposition 7.3.9.2, we obtain a relative version of Corollary 7.2.3.5:

Corollary 7.3.9.4. 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 7.3.9.1, 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 7.2.3.5, $(1')$ is equivalent to the requirement that $\overline{g}$ is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{C}}_{D}$. In particular, the implication $(2') \Rightarrow (1')$ follows from Corollary 7.1.5.20. 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 7.3.9.2). This follows immediately from Corollary 7.2.3.5 (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 7.3.9.2 has a counterpart for the existence of $U$-colimit diagrams.

Proposition 7.3.9.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty$-categories, and suppose we are given a lifting problem

7.45
$$\begin{gathered}\label{equation:relative-colimit-existence-criterion} \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{f_0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleright } \ar [r]^-{g } \ar@ {-->}[ur]^{ \overline{f}_0 } & \operatorname{\mathcal{D}}} \end{gathered}$$

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 (7.45) 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 5.2.2.9. 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 (7.45) 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 7.3.9.1 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 7.3.9.2.

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 7.2.2.9 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 7.3.9.1, we conclude that $\overline{f}_0$ is a $U$-colimit diagram which solves the lifting problem (7.45). $\square$

Corollary 7.3.9.6. 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

7.46
$$\begin{gathered}\label{equation:all-relative-colimits} \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{f} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}} \end{gathered}$$

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 7.3.9.5. 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 7.1.5.20) 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 7.3.9.2). 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 7.1.2.14). $\square$

Corollary 7.3.9.7. 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 7.3.5.5 with Corollary 7.3.9.6. $\square$