Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

7.1.4 Preservation of Limits and Colimits

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Beware that, in general, $F$ need not carry (co)limit diagrams in $\operatorname{\mathcal{C}}$ to (co)limit diagrams in $\operatorname{\mathcal{D}}$. This motivates the following:

Definition 7.1.4.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $K$ be a simplicial set. We will say that $F$ preserves $K$-indexed limits if, for every limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, the composite map $(F \circ \overline{q}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in $\operatorname{\mathcal{D}}$. We will say that $F$ preserves $K$-indexed colimits if, for every colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, the composite map $(F \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Example 7.1.4.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be any functor of $\infty $-categories. Then $F$ preserves $\Delta ^{0}$-indexed limits and colimits. By virtue of Example 7.1.3.15, this is equivalent to the observation that $F$ carries isomorphisms in $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{D}}$ (see Remark 1.4.1.6).

Warning 7.1.4.3. In the formulation of Definition 7.1.4.1, it is not necessary to assume that the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits or colimits. For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category which contains no limit diagrams $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed limits. In practice, we will usually (but not always) apply the terminology of Definition 7.1.4.1 in cases where the $\infty $-category admits $K$-indexed limits or colimits, so that the conclusion of Definition 7.1.4.1 is non-vacuous.

Let us begin with a trivial example.

Proposition 7.1.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $K$ be a simplicial set. Then:

$(1)$

A morphism $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if the composition $F \circ \overline{p}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

$(2)$

A morphism $\overline{p}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if the composition $F \circ \overline{p}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

In particular, the equivalence $F$ preserves $K$-indexed limits and colimits.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram and set $p = \overline{p}|_{K}$. We then have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ / \overline{p} } \ar [r] \ar [d] & \operatorname{\mathcal{D}}_{ / (F \circ \overline{p} )} \ar [d] \\ \operatorname{\mathcal{C}}_{/p} \ar [r] & \operatorname{\mathcal{D}}_{ / (F \circ \overline{p} )}. } \]

Since $F$ is an equivalence of $\infty $-categories, the horizontal maps in this diagram are also equivalences of $\infty $-categories (Corollary 4.6.5.17). It follows that the left vertical map is an equivalence of $\infty $-categories if and only if the right vertical map is an equivalence of $\infty $-categories. The desired result now follows from the criterion of Proposition 7.1.3.18. $\square$

Corollary 7.1.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories and let $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $F \circ \overline{p}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$, then $\overline{p}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof. Set $p = \overline{p}_{K}$, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{D}}$ denote the essential image of $F$ (Definition 4.6.2.9). Since $F \circ \overline{p}$ is a limit diagram, it is final when viewed as an object of the $\infty $-category $\operatorname{\mathcal{D}}_{ / (F \circ p)}$, hence also when viewed as an object of the full subcategory $\operatorname{\mathcal{C}}'_{ / (F \circ p)}$ (Proposition 7.1.1.18). In other words, $F \circ \overline{p}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}'$. Since the functor $F$ is fully faithful, it restricts to an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ (Corollary 4.6.2.19). Applying Proposition 7.1.4.4, we conclude that $\overline{p}$ is a limit diagram in $\operatorname{\mathcal{C}}$. $\square$

Corollary 7.1.4.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:

$(1)$

Let $\overline{p}, \overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Then $\overline{p}$ is a limit diagram if and only if $\overline{q}$ is a limit diagram.

$(2)$

Let $\overline{p}, \overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$. Then $\overline{p}$ is a colimit diagram if and only if $\overline{q}$ is a colimit diagram.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $e: \overline{p} \rightarrow \overline{q}$ be an isomorphism in the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Under the canonical isomorphism

\[ \operatorname{Fun}( \Delta ^1, \operatorname{Fun}( K^{\triangleleft },\operatorname{\mathcal{C}}) ) \simeq \operatorname{Fun}( \Delta ^1 \times K^{\triangleleft }, \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( K^{\triangleleft }, \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \]

we can identify $e$ with a diagram $\overline{r}: K^{\triangleleft } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$, which factors through the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$. Note that the evaluation maps

\[ \operatorname{ev}_0: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]

are trivial Kan fibrations (Corollary 4.4.5.8), and therefore equivalences of $\infty $-categories (Proposition 4.5.2.9). By construction, we have $\overline{p} = \operatorname{ev}_0 \circ \overline{r}$ and $\overline{q} = \operatorname{ev}_1 \circ \overline{r}$. Applying Proposition 7.1.4.4, we deduce that $\overline{p}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $\overline{r}$ is a limit diagram in $\operatorname{Isom}(\operatorname{\mathcal{C}})$. Similarly, $\overline{q}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $\overline{r}$ is a limit diagram in $\operatorname{Isom}(\operatorname{\mathcal{C}})$. $\square$

Corollary 7.1.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and suppose we are given a pair of morphisms $p,q: K \rightarrow \operatorname{\mathcal{C}}$ which are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. Then:

$(1)$

The morphism $p$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ if and only if $q$ can be extended to a limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.

$(2)$

The morphism $p$ can be extended to a colimit diagram $\overline{p}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ if and only if $q$ can be extended to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Suppose that $p$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Since the diagrams $p$ and $q$ are isomorphic, it follows from Corollary 4.4.5.3 that $\overline{p}$ is isomorphic to a diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{q}|_{K} = q$. Applying Corollary 7.1.4.6, we conclude that $\overline{q}$ is also a limit diagram. $\square$

Corollary 7.1.4.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $p: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then:

$(1)$

The morphism $p$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ if and only if the composite map $(F \circ p): K \rightarrow \operatorname{\mathcal{D}}$ can be extended to a limit diagram $K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$.

$(2)$

The morphism $p$ can be extended to a colimit diagram $\overline{p}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ if and only if the composite map $(F \circ p): K \rightarrow \operatorname{\mathcal{D}}$ can be extended to a colimit diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. If $p$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, then Proposition 7.1.4.4 guarantees that $F \circ \overline{p}$ is a colimit diagram in $\operatorname{\mathcal{D}}$ extending $F \circ p$. Conversely, suppose that $F \circ p$ can be extended to a limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an equivalence of $\infty $-categories which is homotopy inverse to $F$, so that $G \circ F$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. Then $(G \circ \overline{q}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ (Proposition 7.1.4.4), and the restriction $(G \circ \overline{q})|_{K} = (G \circ F \circ p)$ is isomorphic to $p$ as an object of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. Applying Corollary 7.1.4.7, we deduce that $p$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. $\square$

Corollary 7.1.4.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are equivalent to one another, and let $K$ be a simplicial set. Then $\operatorname{\mathcal{C}}$ admits $K$-indexed (co)limits if and only if $\operatorname{\mathcal{D}}$ admits $K$-indexed (co)limits.

Remark 7.1.4.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $K$ be a simplicial set, and let $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a limit diagram with restriction $q = \overline{q}|_{K}$. The following conditions are equivalent:

$(1)$

The composition $(F \circ \overline{q}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram.

$(2)$

For every limit diagram $\overline{q}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $\overline{q}'|_{K} = q$, the composition $(F \circ \overline{q}'): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram.

The implication $(2) \Rightarrow (1)$ is immediate. For the converse, we observe that if $\overline{q}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is another limit diagram with $\overline{q}'|_{K} = q$, then $\overline{q}$ and $\overline{q}'$ are isomorphic when viewed as objects of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/q}$, so that $F \circ \overline{q}$ and $F \circ \overline{q}'$ are isomorphic when viewed as objects of the $\infty $-category $\operatorname{\mathcal{D}}_{/ (F \circ q)}$. Since $F \circ \overline{q}$ is a final object of $\operatorname{\mathcal{D}}_{/ (F \circ q)}$, it follows that $F \circ \overline{q}'$ is also a final object of $\operatorname{\mathcal{D}}_{ / (F \circ q)}$ (Corollary 7.1.1.14).

A conservative functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which preserves $K$-indexed limits also reflects them:

Proposition 7.1.4.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $K$ be a simplicial set.

  • Suppose that $\operatorname{\mathcal{C}}$ admits $K$-indexed limits and the functor $F$ preserves $K$-indexed limits. Then a morphism $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $(F \circ \overline{q}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

  • Suppose that $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits and the functor $F$ preserves $K$-indexed colimits. Then a morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if $(F \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Proposition 7.1.4.11 is an immediate consequence of the following more precise assertion:

Lemma 7.1.4.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $q$ can be extended to a limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{q}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is also a limit diagram. Let $\overline{q}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be an arbitrary extension of $q$. Then $\overline{q}'$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $F \circ \overline{q}'$ is a limit diagram in $\operatorname{\mathcal{D}}$.

Proof. Let us identify $\overline{q}$ and $\overline{q}'$ with objects $C$ and $C'$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/q}$. Our assumption that $\overline{q}$ is a limit diagram guarantees that $C$ is a final object of $\operatorname{\mathcal{C}}_{/q}$, so there exists a morphism $u: C' \rightarrow C$ in $\operatorname{\mathcal{C}}_{/q}$. Note that $\overline{q}'$ is a limit diagram if and only if the object $C'$ is also final: that is, if and only if the morphism $u$ is an isomorphism.

Let $v: D' \rightarrow D$ be the image of $u$ under the functor $F_{/q}: \operatorname{\mathcal{C}}_{/q} \rightarrow \operatorname{\mathcal{D}}_{ / (F \circ q)}$. Our assumption that $F \circ \overline{q}$ is a limit diagram guarantees that $D$ is a final object of $\operatorname{\mathcal{D}}_{ / (F \circ q)}$. Consequently, $v$ is an isomorphism if and only if the object $D'$ is also final: that is, if and only if $(F \circ \overline{q}')$ is a limit diagram in $\operatorname{\mathcal{D}}$.

To complete the proof, it will suffice to show that $u$ is an isomorphism in $\operatorname{\mathcal{C}}_{/q}$ if and only if $v = F_{/q}(u)$ is an isomorphism in $\operatorname{\mathcal{D}}_{ / (F \circ q)}$. In fact, the functor $F_{/q}$ is conservative: this follows from our assumption that $F$ is conservative, by virtue of Corollary 4.4.2.11. $\square$

Definition 7.1.4.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $K$ be a simplicial set. We will say that the functor $F$ creates $K$-indexed limits if the following condition is satisfied:

  • Let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ q): K \rightarrow \operatorname{\mathcal{D}}$ admits a limit in $\operatorname{\mathcal{D}}$. Then $q$ can be extended to a limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{q}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

We say that the functor $F$ creates $K$-indexed colimits if it satisfies the following dual condition:

  • Let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ q): K \rightarrow \operatorname{\mathcal{D}}$ admits a colimit in $\operatorname{\mathcal{D}}$. Then $q$ can be extended to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Remark 7.1.4.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $F$ creates $K$-indexed limits and that $F \circ q$ can be extended to a limit diagram $K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. Then an extension $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of $q$ is a limit diagram if and only if $F \circ \overline{q}$ is a limit diagram in $\operatorname{\mathcal{D}}$ (see Lemma 7.1.4.12).

Proposition 7.1.4.15. Let $K$ be a simplicial set, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $K$-indexed limits, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits and the functor $F$ preserves $K$-indexed limits.

$(2)$

The functor $F$ creates $K$-indexed limits.

Proof. The implication $(1) \Rightarrow (2)$ is immediately. Conversely, suppose that $(2)$ is satisfied and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Since $\operatorname{\mathcal{D}}$ admits $K$-indexed limits, $F \circ q$ can be extended to a limit diagram in $\operatorname{\mathcal{D}}$. Since $F$ creates $K$-indexed limits, it follows that there exists a limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $\overline{q}|_{K} = q$ such that $F \circ \overline{q}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Applying Remark 7.1.4.10, we see that this holds for every limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{q}|_{K} =q$, which proves $(1)$. $\square$

The following is an important example of Definition 7.1.4.13:

Proposition 7.1.4.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $A$ be a simplicial set, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then:

$(1)$

The projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ creates $K$-indexed limits, for every simplicial set $K$.

$(2)$

The projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ creates $K$-indexed colimits, for every simplicial set $K$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $K$ be a simplicial set and let $p: K \rightarrow \operatorname{\mathcal{C}}_{f/}$ be a diagram, which we will identify with a morphism of simplicial sets $q: A \star K \rightarrow \operatorname{\mathcal{C}}$ satisfying $q|_{A} = f$. Set $g = q|_{K}$, so that $q$ can also be identified with a diagram $f': A \rightarrow \operatorname{\mathcal{C}}_{/g}$. Suppose that $g$ can be extended to a limit diagram $\overline{g}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Then the projection map $\operatorname{\mathcal{C}}_{/ \overline{g} } \rightarrow \operatorname{\mathcal{C}}_{/g}$ is a trivial Kan fibration (Proposition 7.1.3.18), so that $f'$ can be lifted to a diagram $f'': A \rightarrow \operatorname{\mathcal{C}}_{ / \overline{g} }$. We can then identify $f''$ with a morphism of simplicial sets $\overline{q}: A \star K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ extending $q$, or equivalently with a morphism $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{f/}$ extending $p$. We will complete the proof by showing that $\overline{p}$ is a limit diagram. To prove this, it will suffice to show that $\overline{p}$ is final when regarded as an object of the slice $\infty $-category $(\operatorname{\mathcal{C}}_{f/})_{/p} \simeq (\operatorname{\mathcal{C}}_{/g})_{f'/}$. This follows from Proposition 7.1.1.10, since $\overline{g}$ is a final object of $\operatorname{\mathcal{C}}_{/g}$. $\square$

Corollary 7.1.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $K$ be an arbitrary simplicial set. Then:

$(1)$

If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ admits $K$-indexed limits and the projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed limits.

$(2)$

If $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ admits $K$-indexed colimits and the projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed colimits.

Corollary 7.1.4.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, so that $F$ induces a functor of coslice $\infty $-categories $F': \operatorname{\mathcal{C}}_{u/} \rightarrow \operatorname{\mathcal{D}}_{ (F \circ u)/}$. If the functor $F$ admits a right adjoint, then the functor $F'$ also admits a right adjoint.

Proof. We will show that $F'$ satisfies the criterion of Corollary 7.1.2.14. Fix an object $\overline{Y} \in \operatorname{\mathcal{D}}_{ (F \circ u) / }$, which we identify with a morphism of simplicial sets $K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ carrying the cone point of $K^{\triangleright }$ to an object $Y \in \operatorname{\mathcal{D}}$. We wish to show that the projection map

\[ U: \operatorname{\mathcal{C}}_{u/} \times _{ \operatorname{\mathcal{D}}_{ (F \circ u)/} } ( \operatorname{\mathcal{D}}_{ (F \circ u)/ } )_{/\overline{Y} } \rightarrow \operatorname{\mathcal{C}}_{u/} \]

is a representable right fibration. Unwinding the definitions, we can identify $\overline{Y}$ with a morphism of simplicial sets $\widetilde{u}: K \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y}$, and $U$ with the map $( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )_{\widetilde{u} / } \rightarrow \operatorname{\mathcal{C}}_{u/}$. It will therefore suffice to show that the coslice $\infty $-category $(\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )_{ \widetilde{u} / }$ admits a final object. Since $F$ admits a right adjoint, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y}$ admits a final object (Corollary 7.1.2.14). The desired result now follows from the observation that the projection map

\[ (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )_{ \widetilde{u} / } \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} \]

creates limits (Corollary 7.1.4.17). $\square$

Corollary 7.1.4.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For every simplicial set $K$, the functor $F$ preserves $K$-indexed colimits and the functor $G$ preserves $K$-indexed limits.

Proof. We will show that $F$ preserves $K$-indexed colimits; the assertion that $G$ preserves $K$-indexed limits can be proved by a similar argument. Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, so that $F$ induces a functor $F': \operatorname{\mathcal{C}}_{u/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ u) / }$. We wish to show that the functor $F'$ carries initial objects of $\operatorname{\mathcal{C}}_{u/}$ to initial objects of $\operatorname{\mathcal{D}}_{ (F \circ u)/}$. It follows from Corollary 7.1.4.18 that the functor $F'$ also admits a right adjoint. We may therefore replace $F$ by $F'$ and thereby reduce to the case where $K = \emptyset $. In this case, we must show that if $X$ is an initial object of $\operatorname{\mathcal{C}}$, then $F(X)$ is an initial object of $\operatorname{\mathcal{D}}$. Choose an object $Y \in \operatorname{\mathcal{D}}$; we wish to show that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F( X ), Y )$ is a contractible Kan complex. Proposition 6.2.1.17 supplies a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, G(Y) )$. We conclude by observing that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) )$ is contractible, by virtue of our assumption that the object $X \in \operatorname{\mathcal{C}}$ is initial. $\square$