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7.2.2 Cofinality and Limits

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In ยง7.1.2, we introduced the notion of a limit $\varprojlim (G)$ and colimit $\varinjlim (G)$ for a diagram $G: B \rightarrow \operatorname{\mathcal{C}}$ (Definition 7.1.1.11). Our goal in this section is to show that, if $F: A \rightarrow B$ is a left cofinal morphism of simplicial sets, then the limit $\varprojlim (G)$ (if it exists) can be identified with the limit $\varprojlim (G \circ F)$. Similarly, if $F: A \rightarrow B$ is right cofinal, then the colimit $\varinjlim (G)$ (if it exists) can be identified with the colimit $\varinjlim (G \circ F)$. Our proof is based on the following characterization of (left) cofinality:

Proposition 7.2.2.1. Let $F: A \rightarrow B$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $F$ is left cofinal (in the sense of Definition 7.2.1.1).

$(2)$

The diagram

7.10
\begin{equation} \begin{gathered}\label{equation:cofinality-criterion} \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d]^{F} & A^{\triangleleft } \ar [d]^{ F^{\triangleleft } } \\ B \ar [r] & B^{\triangleleft } } \end{gathered} \end{equation}

is a categorical pushout square of simplicial sets.

$(3)$

The diagram

7.11
\begin{equation} \begin{gathered}\label{equation:cofinality-criterion2} \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d]^{F} & \Delta ^{0} \diamond A \ar [d] \\ B \ar [r] & \Delta ^0 \diamond B } \end{gathered} \end{equation}

is a categorical pushout square (here $\diamond $ denotes the blunt join introduced in Notation 4.5.8.3).

$(4)$

For every $\infty $-category $\operatorname{\mathcal{C}}$ and every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, composition with $F$ induces an equivalence of $\infty $-categories

\[ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} . \]
$(5)$

For every $\infty $-category $\operatorname{\mathcal{C}}$ and every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, the restriction map $\operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{C}}_{ / (G \circ F) }$ is an equivalence of $\infty $-categories.

$(6)$

For every $\infty $-category $\operatorname{\mathcal{C}}$, every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, and every object $X \in \operatorname{\mathcal{C}}$, precomposition with $F$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Hom}_{\operatorname{Fun}(B,\operatorname{\mathcal{C}})}( \underline{X}, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}})}( \underline{X} \circ F, G \circ F); \]

here $\underline{X}: B \rightarrow \operatorname{\mathcal{C}}$ denotes the constant diagram taking the value $X$.

$(7)$

For every $\infty $-category $\operatorname{\mathcal{C}}$, every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, and every object $X \in \operatorname{\mathcal{C}}$, precomposition with $F$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}} (A, \operatorname{\mathcal{C}}_{X/} ). \]

Proof. We first show that $(1)$ implies $(2)$. Let $F: A \rightarrow B$ be a left cofinal morphism of simplicial sets; we wish to show that the diagram (7.10) is a categorical pushout square. By virtue of Corollary 7.2.1.15 (and Proposition 4.5.4.8), we may assume that $F$ is either left anodyne or a trivial Kan fibration. In the second case, the vertical morphisms in the diagram (7.10) are categorical equivalences (see Corollary 4.5.8.9), so the desired result is a special case of Proposition 4.5.4.10. In the second case, Example 4.3.6.5 guarantees that the induced map $B \coprod _{A} A^{\triangleleft } \hookrightarrow B^{\triangleleft }$ is inner anodyne, so the desired result follows from Proposition 4.5.4.11.

Notation 4.5.8.3 supplies a comparison map from the diagram (7.11) to the diagram (7.10), whcih is a levelwise categorical equivalence by virtue of Theorem 4.5.8.8. The equivalence $(2) \Leftrightarrow (3)$ now follows from Proposition 4.5.4.9.

We next show that $(3)$ implies $(4)$. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $G: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram. If condition $(3)$ is satisfied, then the diagram of $\infty $-categories

\[ \xymatrix { \operatorname{Fun}( \Delta ^0 \diamond B, \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^0 \diamond A, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( B, \operatorname{\mathcal{C}}) \ar [r]^{ \circ F} & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \]

is a categorical pullback square. Corollary 4.4.5.3 guaranteees that the vertical maps in this diagram are isofibrations. Invoking Corollary 4.5.2.31 (together with the definition of the blunt join), we deduce that the induced map

\begin{eqnarray*} \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} & \simeq & \operatorname{Fun}( \Delta ^0 \diamond B, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \\ & \rightarrow & \operatorname{Fun}( \Delta ^0 \diamond A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F\} \\ & \simeq & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}( A, \operatorname{\mathcal{C}}) } \{ G \circ F\} \end{eqnarray*}

is an equivalence of $\infty $-categories.

We next prove the equivalences $(4) \Leftrightarrow (5) \Leftrightarrow (6) \Leftrightarrow (7)$. Let $G: B \rightarrow \operatorname{\mathcal{C}}$ be as above. Applying Construction 4.6.4.13, we obtain a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/G} \ar [r] \ar [d]^{\theta } & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \ar [d]^{\theta '} \\ \operatorname{\mathcal{C}}_{ / (G \circ F)} \ar [r] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} , } \]

where the horizontal maps are equivalences of $\infty $-categories (Theorem 4.6.4.17). It follows that $\theta $ is an equivalence of $\infty $-categories if and only if $\theta '$ is an equivalence of $\infty $-categories. This proves the equivalence $(4) \Leftrightarrow (5)$. Note that the functor $\theta '$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \ar [dr] \ar [rr]^{\theta '} & & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} \ar [dl] \\ & \operatorname{\mathcal{C}}, & } \]

where the vertical maps are right fibrations (Corollary 4.6.4.12). Applying Corollary 5.1.7.16 and Proposition 5.1.7.5, we see that $\theta '$ is an equivalence of $\infty $-categories if and only if it induces a homotopy equivalence

\[ \theta '_{X}: \{ \underline{X} \} \operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \rightarrow \{ \underline{X} \circ F \} \operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} \]

for each object $X \in \operatorname{\mathcal{C}}$, which proves the equivalence $(4) \Leftrightarrow (6)$. Unwinding the definitions, we can identify $\theta '$ with the lower horizontal map appearing in the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{C}}_{X/} ) \ar [r]^-{\theta ''_{X}} \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( A, \operatorname{\mathcal{C}}_{X/} ) \ar [d] \\ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( B, \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \ar [r]^-{\theta '_{X}} & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( A, \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}), } \]

where the vertical maps are given by postcomposition with the coslice diagonal morphism $\rho : \operatorname{\mathcal{C}}_{X/} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$. Theorem 4.6.4.17 guarantees that $\rho $ is an equivalence of $\infty $-categories. It is therefore also an an equivalence of left fibrations over $\operatorname{\mathcal{C}}$ (Proposition 5.1.7.5), so that the vertical maps are homotopy equivalences. It follows that $\theta '_{X}$ is a homotopy equivalence if and only if $\theta ''_{X}$ is a homotopy equivalence, which proves the equivalence $(6) \Leftrightarrow (7)$.

We now complete the proof by showing that $(7)$ implies $(1)$. Assume that condition $(7)$ is satisfied; we wish to show that $F$ is left cofinal. Let $q: \widetilde{B} \rightarrow B$ be a left fibration; we must show that composition with $F$ induces a homotopy equivalence $\operatorname{Fun}_{/B}( B, \widetilde{B} ) \rightarrow \operatorname{Fun}_{/B}(A, \widetilde{B} )$. To prove this, we are free to replace $q: \widetilde{B} \rightarrow B$ by any other left fibration which is equivalent to it (in the sense of Definition 5.1.7.1). We may therefore assume without loss of generality that there exists a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{B} \ar [r] \ar [d]^{q} & \operatorname{\mathcal{S}}_{\ast } \ar [d]^{q_{\mathrm{univ}}} \\ B \ar [r]^-{G} & \operatorname{\mathcal{S}}, } \]

where $q_{\mathrm{univ}}: \operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ is the universal left fibration of Corollary 5.6.0.6. We are then reduced to proving that $F$ induces a homotopy equivalence $\operatorname{Fun}_{ / \operatorname{\mathcal{S}}}( B, \operatorname{\mathcal{S}}_{\ast } ) \rightarrow \operatorname{Fun}_{ \operatorname{\mathcal{S}}}( A, \operatorname{\mathcal{S}}_{\ast } )$, which is a special case of $(7)$ (applied to the $\infty $-category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ and the object $X = \Delta ^0$). $\square$

Corollary 7.2.2.2. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a morphism of simplicial sets $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if the composite map

\[ A^{\triangleleft } \xrightarrow {e^{\triangleleft }} B^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

is a $U$-limit diagram.

Proof. Set $f = \overline{f}|_{B}$ and apply Remark 7.1.4.9 to the commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ / f } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ / (f \circ e) } \ar [d] \\ \operatorname{\mathcal{D}}_{ / (U \circ f) } \ar [r] & \operatorname{\mathcal{D}}_{ / (U \circ f \circ e) }, } \]

noting that the horizontal maps are equivalences by virtue of Proposition 7.2.2.1. $\square$

Corollary 7.2.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a morphism of simplicial sets $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if the composite map

\[ A^{\triangleleft } \xrightarrow {e^{\triangleleft }} B^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

is a limit diagram.

Proof. Apply Corollary 7.2.2.2 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$ (see Example 7.1.5.3). $\square$

Remark 7.2.2.4. The converse of Corollary 7.2.2.3 is also true: if $e: A \rightarrow B$ is a morphism of simplicial sets having the property that precomposition with the induced map $e^{\triangleleft }: A^{\triangleleft } \rightarrow B^{\triangleleft }$ carries limit diagrams to limit diagrams, then $e$ is left cofinal. Moreover, it suffices check this condition for diagrams in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (see Corollary 7.4.5.14).

Corollary 7.2.2.5. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories and let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$. Then:

  • If $Y$ is an initial object of $\operatorname{\mathcal{C}}$, then a diagram $\operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-limit diagram if and only if it carries $\{ Y\} ^{\triangleleft } \simeq \Delta ^1$ to a $U$-cartesian morphism of $\operatorname{\mathcal{D}}$.

  • If $Y$ is a final object of $\operatorname{\mathcal{K}}$, then a diagram $\operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram if and only if it carries $\{ Y\} ^{\triangleright } \simeq \Delta ^1$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.

Proof. If $Y$ is an initial object of $\operatorname{\mathcal{C}}$, then the inclusion map $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is left cofinal (Corollary 4.6.7.24). The first assertion now follows by combining Corollary 7.2.2.2 with Example 7.1.5.9. The second assertion follows by a similar argument. $\square$

Corollary 7.2.2.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Then:

  • If $\operatorname{\mathcal{C}}$ has an initial object $Y$, then a functor $\operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram if and only if it carries $\{ Y\} ^{\triangleleft } \simeq \Delta ^1$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

  • If $\operatorname{\mathcal{C}}$ has a final object $Y$, then a functor $\operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram if and only if it carries $\{ Y\} ^{\triangleright } \simeq \Delta ^1$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

Proof. Apply Corollary 7.2.2.5 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$ (and use Example 5.1.1.4). $\square$

Corollary 7.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $C \in \operatorname{\mathcal{C}}$ and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Suppose we are given a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : \underline{C} \rightarrow f$, where $\underline{C} \in \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ denotes the constant diagram taking the value $C$. Then $\alpha $ exhibits $C$ as a limit of $f$ (in the sense of Definition 7.1.1.1) if and only if the induced natural transformation $\alpha |_{A}: \underline{C}|_{A} \rightarrow f|_{A}$ exhibits $C$ as a limit of the diagram $f|_{A}$.

Proof. By virtue of Remark 7.1.1.7, we are free to modify the natural transformation $\alpha $ by a homotopy and may therefore assume that it corresponds to a morphism of simplicial sets $\Delta ^0 \diamond B \rightarrow \operatorname{\mathcal{C}}$ which factors through the categorical equivalence $\Delta ^0 \diamond B \twoheadrightarrow \Delta ^0 \star B$ of Theorem 4.5.8.8. In this case, the desired result follows from Corollary 7.2.2.3 and Remark 7.1.2.6. $\square$

Corollary 7.2.2.8. Let $e: A \rightarrow B$ be a morphism of simplicial sets and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $e$ is left cofinal and the functor $F$ preserves $A$-indexed limits, then $F$ preserves $B$-indexed limits. If $e$ is right cofinal and the functor $F$ preserves $A$-indexed colimits, then the functor $F$ preserves $B$-indexed colimits.

Proposition 7.2.2.9. Suppose we are given a commutative diagram of simplicial sets

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

where $U$ is an inner fibration of $\infty $-categories. Let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. The following conditions are equivalent:

$(1)$

There exists a $U$-limit diagram $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}|_{B} = f$ and $U \circ \overline{f} = \overline{g}$.

$(2)$

There exists a $U$-limit diagram $\overline{f}_0: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}_0|_{A} = f \circ e$ and $U \circ \overline{f}_0 = \overline{g} \circ e^{\triangleleft }$.

Proof. The implication $(1) \Rightarrow (2)$ follows by observing that if $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram, then the left cofinality of $e$ guarantees that $\overline{f} \circ e^{\triangleleft }$ is also a $U$-limit diagram (Corollary 7.2.2.2). We will complete the proof by showing that $(2)$ implies $(1)$. By virtue of Corollary 7.2.1.15, we can assume that the morphism $e$ is either left anodyne or a trivial Kan fibration. We first treat the case where $e$ is a trivial Kan fibration. Let $s: B \rightarrow A$ be a section of $e$, and let $\overline{f}_0: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfy the requirements of $(2)$. Let $\overline{f}$ denote the composite map

\[ B^{\triangleleft } \xrightarrow { s^{\triangleleft } } A^{\triangleleft } \xrightarrow { \overline{f}_0 } \operatorname{\mathcal{C}}. \]

It follows immediately from the construction that $\overline{f}|_{B} = f$ and $U \circ \overline{f} = \overline{g}$. Moreover, the composition $\overline{f} \circ e^{\triangleleft }$ is isomorphic to $\overline{f}_0$ (as an object of the $\infty $-category $\operatorname{Fun}( B^{\triangleleft }, \operatorname{\mathcal{C}})$), and is therefore also a $U$-limit diagram (Proposition 7.1.5.13). Since $e$ is left cofinal, it follows that $\overline{f}$ is also a $U$-limit diagram (Corollary 7.2.2.2).

We now treat the case where $e$ is left anodyne. In this case, the induced map $A^{\triangleleft } {\coprod }_{A} B \hookrightarrow B^{\triangleleft }$ is inner anodyne. Since $U$ is an inner fibration, we can extend $f$ to a morphism $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ \overline{f} = \overline{g}$ and $\overline{f} \circ e^{\triangleleft } = \overline{f}_0$. Since $e$ is left cofinal, the morphism $\overline{f}$ is automatically a $U$-limit diagram (Corollary 7.2.2.2). $\square$

Corollary 7.2.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ has a limit if and only if the composite diagram $(f \circ e): A \rightarrow \operatorname{\mathcal{C}}$ has a limit.

Proof. If $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f$, then Corollary 7.2.2.3 guarantees that $\overline{f} \circ e^{\triangleleft }: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f \circ e$. Conversely, if $f \circ e$ can be extended to a colimit diagram, then Proposition 7.2.2.9 (applied in the special case $\operatorname{\mathcal{D}}= \Delta ^0$) guarantees that $f$ can also be extended to a colimit diagram. $\square$

Corollary 7.2.2.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets, and let $f: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then an object $X \in \operatorname{\mathcal{C}}$ is a limit of $f$ if and only if it is a limit of the diagram $(f \circ e): A \rightarrow \operatorname{\mathcal{C}}$.

Proof. If an object $X \in \operatorname{\mathcal{C}}$ is a limit of $f$, then we can choose a limit diagram $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ carrying the cone point of $f^{\triangleleft }$ to the object $X$. Applying Corollary 7.2.2.10, we deduce that $\overline{f} \circ e^{\triangleleft }$ exhibits $X$ as a limit of the diagram $f \circ e$. Conversely, if $X$ is a limit of the diagram $f \circ e$, then Corollary 7.2.2.10 guarantees that the diagram $f$ admits a limit $Y \in \operatorname{\mathcal{C}}$. The preceding argument shows that $Y$ is also a limit of the diagram $f \circ e$. Applying Proposition 7.1.1.12, we deduce that $Y$ is isomorphic to $X$, so that $X$ is also a limit of the diagram $f$. $\square$

Corollary 7.2.2.12. Let $e: A \rightarrow B$ be a morphism of simplicial sets and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $e$ is left cofinal and $\operatorname{\mathcal{C}}$ admits $A$-indexed limits, then $\operatorname{\mathcal{C}}$ also admits $B$-indexed limits. If $e$ is right cofinal and $\operatorname{\mathcal{C}}$ admits $A$-indexed colimits, then $\operatorname{\mathcal{C}}$ also admits $B$-indexed colimits.

Corollary 7.2.2.13. Let $e: A \rightarrow B$ be a morphism of simplicial sets and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $e$ is left cofinal and the functor $F$ creates $A$-indexed limits, then $F$ creates $B$-indexed limits. If $e$ is right cofinal and the functor $F$ creates $A$-indexed colimits, then the functor $F$ creates $B$-indexed colimits.

Corollary 7.2.2.14. Suppose we are given lifting problem

7.12
\begin{equation} \begin{gathered}\label{equation:easy-relative-colimit-existence} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ f } \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}^{\triangleleft } \ar [r] \ar@ {-->}[ur]^{\overline{f}} & \operatorname{\mathcal{E}}, } \end{gathered} \end{equation}

where $\operatorname{\mathcal{C}}$ is an $\infty $-category and $U$ is a cartesian fibration of $\infty $-categories. If $\operatorname{\mathcal{C}}$ has a final object $C$, then (7.12) admits a solution $\overline{f}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-limit diagram.

Proof. Using Proposition 7.2.2.9 and Corollary 4.6.7.24, we can replace $\operatorname{\mathcal{C}}$ by the simplicial set $\{ C\} \simeq \Delta ^0$, in which case the desired result follows from our assumption that $U$ is a cartesian fibration (see Example 7.1.5.9). $\square$