# Kerodon

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### 7.1.5 Relative Limits and Colimits

We now introduce a relative version of Definition 7.1.2.4.

Definition 7.1.5.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with restriction $f = \overline{f}|_{K}$, so that $U$ induces a functor $U_{/f}: \operatorname{\mathcal{C}}_{ / f } \rightarrow \operatorname{\mathcal{D}}_{ / (U \circ f) }$. We will say that $\overline{f}$ is a $U$-limit diagram if it is $U_{/f}$-final when viewed as an object of the $\infty$-category $\operatorname{\mathcal{C}}_{/f}$. Similarly, we say that a morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ with restriction $g = \overline{g}|_{K}$ is a $U$-colimit diagram if $\overline{g}$ is $U_{g/}$-initial when viewed as an object of the $\infty$-category $\operatorname{\mathcal{C}}_{g/}$, where $U_{/g}: \operatorname{\mathcal{C}}_{g/} \rightarrow \operatorname{\mathcal{D}}_{ (U \circ g)/ }$ denotes the functor induced by $U$.

Remark 7.1.5.2. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories. Then a morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if the opposite map $\overline{f}^{\operatorname{op}}: (K^{\operatorname{op}})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is an $U^{\operatorname{op}}$-colimit diagram.

Example 7.1.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and $U: \operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$ be the projection map. Then a morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram (in the sense of Definition 7.1.5.1) if and only if it is a limit diagram (in the sense of Definition 7.1.2.4). Similarly, a morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if it is a colimit diagram.

Example 7.1.5.4. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty$-categories. Then every morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram, and every morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram. This follows by combining Example 7.1.4.4 with Corollary 4.6.4.20.

Example 7.1.5.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then an object $C \in \operatorname{\mathcal{C}}$ is $U$-final if and only if it is a $U$-limit diagram when viewed as a morphism of simplicial sets $(\emptyset )^{\triangleleft } \simeq \Delta ^0 \rightarrow \operatorname{\mathcal{C}}$. Similarly, $C$ is $U$-initial if and only if it is a $U$-colimit diagram when viewed as a morphism of simplicial sets $(\emptyset )^{\triangleright } \simeq \Delta ^0 \rightarrow \operatorname{\mathcal{C}}$.

Remark 7.1.5.6. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{U} \ar [r]^-{F} & \operatorname{\mathcal{C}}' \ar [d]^{U'} \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}', }$

where the horizontal maps are equivalences of $\infty$-categories. Then a morphism of simplicial sets $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if $F \circ \overline{f}$ is a $U'$-limit diagram. Similarly, a morphism of simplicial sets $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if $F \circ \overline{g}$ is a $U'$-colimit diagram. This follows by combining Remark 7.1.4.9 with Corollary 4.6.4.19.

Remark 7.1.5.7. Let $U_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, and let $U_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which is isomorphic to $U_0$ (as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then a diagram $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U_0$-limit diagram if and only if it is a $U_1$-limit diagram (see Remark 7.1.4.8). This follows by applying Remark 7.1.5.6 to each square of the diagram

$\xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}\ar [d]^{ U_0 } & \operatorname{\mathcal{C}}\ar [d]^{ U } \ar [l]_{\operatorname{id}} \ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}\ar [d]^{ U_1 } \\ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{D}}) & \operatorname{Isom}(\operatorname{\mathcal{D}}) \ar [l]_{ \operatorname{ev}_0 } \ar [r]^-{ \operatorname{ev}_{1} } & \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{D}}), }$

where $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{Isom}(\operatorname{\mathcal{D}})$ classifies an isomorphism between $U_0$ and $U_1$; note that $\operatorname{ev}_0$ and $\operatorname{ev}_1$ are trivial Kan fibrations by virtue of Corollary 4.4.5.10.

Remark 7.1.5.8. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism, and set $f = \overline{f}|_{K}$, so that $U$ induces a functor

$U': \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f} \times _{ \operatorname{\mathcal{D}}_{/(U \circ f)}} \operatorname{\mathcal{D}}_{ / (U \circ \overline{f} ) }.$

By virtue of Proposition 7.1.4.16, the following conditions are equivalent:

$(1)$

The morphism $\overline{f}$ is a $U$-limit diagram.

$(2)$

The functor $U'$ is an equivalence of $\infty$-categories.

If $U$ is an inner fibration of $\infty$-categories, then the functor $U'$ is automatically a right fibration (Proposition 4.3.6.8). In this case, we can replace $(1)$ and $(2)$ by either of the following conditions:

$(3)$

The functor $U'$ is a trivial Kan fibration.

$(4)$

Each fiber of $U'$ is a contractible Kan complex.

The equivalence of $(2) \Leftrightarrow (3)$ follows from Proposition 4.5.5.20, and the equivalence $(3) \Leftrightarrow (4)$ from Proposition 4.4.2.14.

Example 7.1.5.9. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories. Then:

• A morphism $e$ of $\operatorname{\mathcal{C}}$ is $U$-cartesian (in the sense of Definition 5.1.1.1) if and only if it is a $U$-limit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.

• A morphism $f$ of $\operatorname{\mathcal{C}}$ is $U$-cocartesian (in the sense of Definition 5.1.1.1) if and only if it is a $U$-colimit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

This follows by combining Remark 7.1.5.8 with Proposition 5.1.1.13.

Example 7.1.5.10. Let $K$ be a weakly contractible simplicial set and let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration of $\infty$-categories. Then every morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram (see Proposition 4.3.7.6). Similarly, if $U$ is a left fibration, then every morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram.

Remark 7.1.5.11. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories and let $K$ be a simplicial set. Using Remark 7.1.5.8, we see that a morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if every lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \star K \ar [r]^-{\rho } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \Delta ^{n} \star K \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}}$

admits a solution, provided that $n \geq 1$ and the the restriction of $\rho$ to $\{ n\} \star K \simeq K^{\triangleleft }$ coincides with $\overline{f}$.

Proposition 7.1.5.12. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Then $\overline{f}$ is a $U$-limit diagram if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the diagram of morphism spaces

7.1
$$\begin{gathered}\label{equation:U-colimit-by-morphism-space} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \underline{C}, \overline{f} ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( \underline{C}|_{K}, \overline{f}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{D}}) }( U \circ \underline{C}, U \circ \overline{f} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ \underline{C}|_{K}, U \circ \overline{f}|_{K} ) } \end{gathered}$$

is a homotopy pullback square; here we let $\underline{C} \in \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$ denote the constant diagram taking the value $C$.

Proof. Set $f = \overline{f}|_{K}$. Note that the restriction maps

$\operatorname{\mathcal{C}}_{/\overline{f}} \rightarrow \operatorname{\mathcal{C}}_{/f } \quad \quad \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{\mathcal{D}}_{ / (U \circ \overline{f} )} \rightarrow \operatorname{\mathcal{D}}_{/(U \circ f) }$

are right fibrations of simplicial sets (Corollary 4.3.6.11). It follows that we can regard the map

$U': \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f} \times _{ \operatorname{\mathcal{D}}_{/(U \circ f)}} \operatorname{\mathcal{D}}_{ / (U \circ \overline{f} ) }$

of Remark 7.1.5.8 as a functor between $\infty$-categories which are right-fibered over $\operatorname{\mathcal{C}}$. Combining Remark 7.1.5.8 with the criterion of Corollary 5.1.6.4, we see that $\overline{f}$ is a $U$-colimit diagram if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the induced map

$U'_{C}: \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f} \times _{ \operatorname{\mathcal{D}}_{/(U \circ f)}} \operatorname{\mathcal{D}}_{ / (U \circ \overline{f} ) }$

is a homotopy equivalence of Kan complexes.

To complete the proof, it will suffice to show that $U'_{C}$ is a homotopy equivalence if and only if the diagram (7.1) is a homotopy pullback square. To see this, we note that Proposition 4.6.5.10 supplies a levelwise homotopy equivalence of (7.1) with the diagram

7.2
$$\begin{gathered}\label{equation:U-colimit-by-morphism-space2} \xymatrix@R =50pt@C=50pt{ \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{ / \overline{f} } \ar [r] \ar [d] & \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f} \ar [d] \\ \{ U(C) \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / (U \circ \overline{f} )} \ar [r] & \{ U(C) \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / (U \circ f) }. } \end{gathered}$$

It will therefore suffice to show that (7.2) is a homotopy pullback square if and only if $U'_{C}$ is a homotopy equivalence (Corollary 3.4.1.12). This is a special case of Example 3.4.1.3, since the horizontal maps in the diagram (7.2) are Kan fibrations (combine Corollaries 4.3.6.11 and 4.4.3.8). $\square$

Proposition 7.1.5.13. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $\overline{u}, \overline{v}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be diagrams which are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a $U$-limit diagram if and only if $\overline{v}$ is a $U$-limit diagram.

Proof. We proceed as in the proof of Corollary 7.1.2.14. Let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$, and define $\operatorname{Isom}(\operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})$ similarly. For $i \in \{ 0,1\}$, the evaluation functors

$\operatorname{ev}_ i: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{ev}_ i: \operatorname{Isom}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$

are trivial Kan fibrations (Corollary 4.4.5.10), and therefore equivalences of $\infty$-categories (Proposition 4.5.3.11). Our assumption that $\overline{u}$ and $\overline{v}$ are isomorphic guarantees that we can choose a diagram $\overline{w}: K^{\triangleleft } \rightarrow \operatorname{Isom}(\operatorname{\mathcal{C}})$ satisfying $\operatorname{ev}_0 \circ \overline{w} = \overline{u}$ and $\operatorname{ev}_1 \circ \overline{w} = \overline{v}$. Applying Remark 7.1.5.6 to the commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Isom}(\operatorname{\mathcal{C}}) \ar [r]^-{ \operatorname{ev}_0 } \ar [d]^{U'} & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{Isom}(\operatorname{\mathcal{D}}) \ar [r]^-{ \operatorname{ev}_0 } & \operatorname{\mathcal{D}}, }$

we see that $\overline{u}$ is a $U$-limit diagram if and only if $\overline{w}$ is a $U'$-limit diagram. A similar argument shows that this is equivalent to the requirement that $\overline{v}$ is a $U$-limit diagram. $\square$

Proposition 7.1.5.14 (Transitivity). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories.

$(1)$

Let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{f}$ is a $V$-limit diagram. Then $\overline{f}$ is a $U$-limit diagram if and only if it is a $(V \circ U)$-limit diagram.

$(2)$

Let $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{g}$ is a $V$-colimit diagram. Then $\overline{g}$ is a $U$-colimit diagram if and only if it is a $(V \circ U)$-colimit diagram.

Proof. Apply Remark 7.1.4.6. $\square$

Corollary 7.1.5.15. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, where $V$ is fully faithful. Then:

$(1)$

A morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if it is a $(V \circ U)$-limit diagram.

$(2)$

A morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if it is a $(V \circ U)$-colimit diagram.

Corollary 7.1.5.16. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then:

$(1)$

Let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{f}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a $U$-limit diagram.

$(2)$

Let $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{g}$ is a colimit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{g}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a $U$-colimit diagram.

Proof. Apply Proposition 7.1.5.14 in the case $\operatorname{\mathcal{E}}= \Delta ^{0}$ (and use Example 7.1.5.3). $\square$

Corollary 7.1.5.17. Let $K$ be a weakly contractible simplicial set and let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. If $U$ is a left fibration, then it creates $K$-indexed colimits. If $U$ is a right fibration, then it creates $K$-indexed limits.

Proof. Assume $U$ is a right fibration; we will show that it creates $K$-indexed limits (the analogous statement for left fibrations follows by a similar argument). Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram and suppose that $U \circ f$ can be extended to a limit diagram $g: K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. Since the inclusion $K \hookrightarrow K^{\triangleleft }$ is right anodyne (Example 4.3.7.10), our assumption that $U$ is a right fibration guarantees that the lifting problem

$\xymatrix@R =50pt@C=50pt{ K \ar [d] \ar [r]^-{f} & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleleft } \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}}$

has a solution. Since $K$ is weakly contractible, the morphism $\overline{f}$ is automatically a $U$-limit diagram (Example 7.1.5.10). Applying Corollary 7.1.5.16, we see that $\overline{f}$ is a limit diagram. $\square$

Corollary 7.1.5.18. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $K$ be a weakly contractible simplicial set. Then:

• If $U$ is a right fibration and the $\infty$-category $\operatorname{\mathcal{D}}$ admits $K$-indexed limits, then $\operatorname{\mathcal{C}}$ also admits $K$-indexed limits and $U$ preserves $K$-indexed limits.

• If $U$ is a left fibration and the $\infty$-category $\operatorname{\mathcal{D}}$ admits $K$-indexed colimits, then $\operatorname{\mathcal{C}}$ also admits $K$-indexed colimits and $U$ preserves $K$-indexed colimits.

Proposition 7.1.5.19 (Base Change). Suppose we are given a commutative diagram of $\infty$-categories

7.3
$$\begin{gathered}\label{equation:base-change-relative-limit5} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [rr]^{F'} \ar [dr]_{U'} \ar [dd]^{G} & & \operatorname{\mathcal{D}}' \ar [dl] \ar [dd] \\ & \operatorname{\mathcal{E}}' \ar [dd] & \\ \operatorname{\mathcal{C}}\ar [dr]_{ U } \ar [rr]^(.4){F} & & \operatorname{\mathcal{D}}\ar [dl]^{V} \\ & \operatorname{\mathcal{E}}& } \end{gathered}$$

where each square is a pullback and the diagonal maps are inner fibrations. Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}'$ be a morphism of simplicial sets. Then:

$(1)$

If $G \circ \overline{f}$ is an $F$-colimit diagram in the $\infty$-category $\operatorname{\mathcal{C}}$, then $\overline{f}$ is an $F'$-colimit diagram in the $\infty$-category $\operatorname{\mathcal{C}}'$.

$(2)$

Assume that $U$ and $V$ are cartesian fibrations, and that the functor $F$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{D}}$. If $\overline{f}$ is an $F'$-colimit diagram in the $\infty$-category $\operatorname{\mathcal{C}}'$, then $G \circ \overline{f}$ is an $F$-colimit diagram in the $\infty$-category $\operatorname{\mathcal{C}}$.

Proof. Set $f = \overline{f}|_{K}$. By virtue of Corollary 4.3.6.10 and Proposition 5.1.4.19, we can replace (7.3) by the commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'_{f/} \ar [rr] \ar [dr] \ar [dd] & & \operatorname{\mathcal{D}}'_{ (F' \circ f)/} \ar [dl] \ar [dd] \\ & \operatorname{\mathcal{E}}'_{ (U' \circ f)/} \ar [dd] & \\ \operatorname{\mathcal{C}}_{ (G \circ f)/ } \ar [rr] \ar [dr] & & \operatorname{\mathcal{D}}_{ (F \circ G \circ f)/} \ar [dl] \\ & \operatorname{\mathcal{E}}_{ (U \circ G \circ f)/} & }$

and thereby reduce to the special case $K = \emptyset$. In this case, the desired result follows from Proposition 7.1.4.19. $\square$

Corollary 7.1.5.20. 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 let

$\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$

be a diagram. If $\overline{f}$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$, then it is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{C}}_{D}$. The converse holds if $U$ is a cartesian fibration.

Proof. Apply Proposition 7.1.5.19 in the special case $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}' = \{ D\}$. $\square$

Remark 7.1.5.21. Corollary 7.1.5.20 has an obvious counterpart for $U$-limit diagrams under the assumption that $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration, which can be proved in the same way. It also has a more subtle counterpart for $U$-colimit diagrams when $U$ is a cocartesian fibration (or $U$-limit diagrams when $U$ is a cartesian fibration), which we will discuss in ยง7.3.9 (see Proposition 7.3.9.2).

Beware that the conclusion of Corollary 7.1.5.20 does not necessarily hold if $U$ is not a cartesian fibration. However, we have the following slightly weaker result:

Proposition 7.1.5.22. 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 let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ be a diagram. Then $\overline{f}$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it satisfies the following condition:

$(\ast )$

For every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the morphism

$K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}_{D} \hookrightarrow \Delta ^{1} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$

is a $U'$-colimit diagram, where $U': \Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ is given by projection onto the first factor.

Proof. Assume that condition $(\ast )$ is satisfied; we will show that $\overline{f}$ is a $U$-colimit diagram (the converse follows from Proposition 7.1.5.19). Set $f = \overline{f}|_{K}$. By virtue of Proposition 7.1.5.12, it will suffice to show that for each vertex $C \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

7.4
$$\begin{gathered}\label{equation:relative-colimit-by-fiber-preliminary} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \overline{f}, \underline{C}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f, \underline{C}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ \overline{f}, U \circ \underline{C} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C}|_{K}) } \end{gathered}$$

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 (7.4) 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 $u \in \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ \overline{f}, U \circ \underline{C} )$, the induced map

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

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 $\operatorname{\mathcal{D}}$. The desired result now follows from $(\ast )$. $\square$

Remark 7.1.5.23. In the situation of Proposition 7.1.5.22, we can replace condition $(\ast )$ by the following:

$(\ast ')$

For every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the morphism

$K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}_{D} \hookrightarrow \Delta ^{1} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$

is a colimit diagram in the $\infty$-category $\Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$.

See Corollary 7.1.5.16.