# Kerodon

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

### 7.1.3 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 7.1.3.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a 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 $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. We say that the limit of $q$ is preserved by $F$ if the composition $F \circ \overline{q}$ is a limit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$. Similarly, if $q$ can be extended to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, we say that the colimit of $q$ is preserved by $F$ if $F \circ \overline{q}$ is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$.

Remark 7.1.3.2. In the situation of Definition 7.1.3.1, the condition that $F$ preserves the (co)limit of a diagram $q: K \rightarrow \operatorname{\mathcal{C}}$ depends only on the diagram $q$, and not on the extension $\overline{q}$ (see Corollary 7.1.2.14).

Remark 7.1.3.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram which admits a limit in $\operatorname{\mathcal{C}}$. Choose an object $X \in \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : \underline{X} \rightarrow q$ which exhibits $X$ as a limit of $q$. Then $F$ preserves the limit of $q$ if and only if the natural transformation $F(\alpha )$ exhibits the object $F(X)$ as a limit of the diagram $F \circ q$.

Definition 7.1.3.4. 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 7.1.3.5. 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 7.1.2.10, 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 7.1.3.6. In the formulation of Definition 7.1.3.4, 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 7.1.3.4 in cases where the $\infty$-category admits $K$-indexed limits or colimits, so that the conclusion of Definition 7.1.3.4 is non-vacuous.

Exercise 7.1.3.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $K$ be a simplicial set. Show that $F$ preserves $K$-indexed limits if and only if it satisfies the following condition:

• For every diagram $u: K \rightarrow \operatorname{\mathcal{C}}$ and every natural transformation $\alpha : \underline{Y} \rightarrow u$ which exhibits an object $Y \in \operatorname{\mathcal{C}}$ as a limit of $u$ (in the sense of Definition 7.1.1.1), the image $F(\alpha ): \underline{F(Y)} \rightarrow (F \circ u)$ exhibits the object $F(Y) \in \operatorname{\mathcal{D}}$ as a limit of the diagram $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$.

Variant 7.1.3.8. It will often be useful to extend the terminology of Definition 7.1.3.4, replacing the individual simplicial set $K$ by a collection of simplicial sets.

• We say that a functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves finite limits if it preserves $K$-indexed limits, for every finite simplicial set $K$.

• We say that a functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves finite colimits if it preserves $K$-indexed colimits, for every finite simplicial set $K$.

• We say that a functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves small limits if it preserves $K$-indexed limits, for every small simplicial set $K$.

• We say that a functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves small colimits if it preserves $K$-indexed colimits, for every small simplicial set $K$.

Let us begin with a trivial example.

Proposition 7.1.3.9. 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 $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if the composition $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

$(2)$

A morphism $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if the composition $F \circ \overline{u}$ 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 $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram and set $u = \overline{u}|_{K}$. We then have a commutative diagram of $\infty$-categories

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

Since $F$ is an equivalence of $\infty$-categories, the horizontal maps in this diagram are also equivalences of $\infty$-categories (Corollary 4.6.4.19). 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 7.1.2.12. $\square$

Variant 7.1.3.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty$-categories and let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $F \circ \overline{u}$ is a limit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$, then $\overline{u}$ is a limit diagram in the $\infty$-category $\operatorname{\mathcal{C}}$.

Corollary 7.1.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty$-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then:

$(1)$

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

$(2)$

The morphism $u$ can be extended to a colimit diagram $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ if and only if the composite map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ 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 $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, then Proposition 7.1.3.9 guarantees that $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$ extending $F \circ u$. Conversely, suppose that $F \circ u$ can be extended to a limit diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. 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 $(G \circ \overline{v}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ (Proposition 7.1.3.9), and the restriction $(G \circ \overline{v})|_{K} = (G \circ F \circ u)$ is isomorphic to $u$ as an object of the $\infty$-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. Applying Corollary 7.1.2.15, we deduce that $u$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. $\square$

Corollary 7.1.3.12. 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 7.1.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $K$ be a simplicial set, and let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a limit diagram with restriction $u = \overline{u}|_{K}$. The following conditions are equivalent:

$(1)$

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

$(2)$

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

The implication $(2) \Rightarrow (1)$ is immediate. For the converse, we observe that if $\overline{u}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is another limit diagram with $\overline{u}'|_{K} = u$, then $\overline{u}$ and $\overline{u}'$ are isomorphic when viewed as objects of the slice $\infty$-category $\operatorname{\mathcal{C}}_{/u}$, so that $F \circ \overline{u}$ and $F \circ \overline{u}'$ are isomorphic when viewed as objects of the $\infty$-category $\operatorname{\mathcal{D}}_{/ (F \circ u)}$. Since $F \circ \overline{u}$ is a final object of $\operatorname{\mathcal{D}}_{/ (F \circ u)}$, it follows that $F \circ \overline{u}'$ is also a final object of $\operatorname{\mathcal{D}}_{ / (F \circ u)}$ (Corollary 4.6.6.16).

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

Proposition 7.1.3.14. 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 $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ 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 $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if $(F \circ \overline{u}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Proposition 7.1.3.14 is an immediate consequence of the following more precise assertion:

Lemma 7.1.3.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty$-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is also a limit diagram. Let $\overline{u}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be an arbitrary extension of $u$. Then $\overline{u}'$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $F \circ \overline{u}'$ is a limit diagram in $\operatorname{\mathcal{D}}$.

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

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

To complete the proof, it will suffice to show that $f$ is an isomorphism in $\operatorname{\mathcal{C}}_{/u}$ if and only if $g = F_{/u}(f)$ 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 4.4.2.12. $\square$

Definition 7.1.3.16. 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 $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ admits a limit in $\operatorname{\mathcal{D}}$. Then $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ 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 $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ admits a colimit in $\operatorname{\mathcal{D}}$. Then $u$ can be extended to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Remark 7.1.3.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty$-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $F$ creates $K$-indexed limits and that $F \circ u$ can be extended to a limit diagram $K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. Then an extension $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of $u$ is a limit diagram if and only if $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$ (see Lemma 7.1.3.15).

Proposition 7.1.3.18. 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 $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Since $\operatorname{\mathcal{D}}$ admits $K$-indexed limits, $F \circ u$ 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 $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $\overline{u}|_{K} = u$ such that $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Applying Remark 7.1.3.13, we see that this holds for every limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{u}|_{K} =u$, which proves $(1)$. $\square$

The following is an important example of Definition 7.1.3.16:

Proposition 7.1.3.19. 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 7.1.2.12), 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 4.6.6.13, since $\overline{g}$ is a final object of $\operatorname{\mathcal{C}}_{/g}$. $\square$

Corollary 7.1.3.20. 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 7.1.3.21. 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 6.2.4.6 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. $\square$

Corollary 7.1.3.22 (Limits in a Reflective Localization). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory (Definition 6.2.2.1). Then a diagram $u: K \rightarrow \operatorname{\mathcal{C}}_0$ admits a limit in $\operatorname{\mathcal{C}}_0$ if and only if it admits a limit in $\operatorname{\mathcal{C}}$. In this case, the limit of $u$ is preserved by the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$.

Proof. By virtue of Proposition 6.2.2.8, the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint. Invoking Corollary 7.1.3.21, we deduce that if $u$ admits a limit in $\operatorname{\mathcal{C}}_0$, then that limit is preserved by the inclusion functor (and therefore $u$ admits a limit in $\operatorname{\mathcal{C}}$). For the converse, assume that $u$ admits a limit $C$ in the $\infty$-category $\operatorname{\mathcal{C}}$. We will complete the proof by showing that $C$ is isomorphic to an object $C' \in \operatorname{\mathcal{C}}_0$; in this case, $C'$ is also a limit of $u$ in the $\infty$-category $\operatorname{\mathcal{C}}$ (Proposition 7.1.1.12), and therefore also a limit of $u$ in the $\infty$-category $\operatorname{\mathcal{C}}_0$ (Remark 7.1.1.10). Replacing $\operatorname{\mathcal{C}}_0$ by its essential image, we may assume that $\operatorname{\mathcal{C}}_0$ is replete. It follows that $\operatorname{\mathcal{C}}_0$ is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects, for some collection of morphisms $W$ of $\operatorname{\mathcal{C}}$ (Proposition 6.3.3.9). We are therefore reduced to showing that the object $C = \varprojlim (u)$ is $W$-local, which follows from Proposition 7.1.1.14. $\square$

Variant 7.1.3.23 (Colimits in a Reflective Localization). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $u: K \rightarrow \operatorname{\mathcal{C}}_0$ be a diagram. If $u$ admits a colimit in $\operatorname{\mathcal{C}}$, then it also admits a colimit in $\operatorname{\mathcal{C}}_0$.

Proof. By virtue of Proposition 6.2.2.8, the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$. If $u$ admits a colimit in $\operatorname{\mathcal{C}}$, then $L \circ u$ admits a colimit in $\operatorname{\mathcal{C}}_0$ (Corollary 7.1.3.21). Since $u$ factors through $\operatorname{\mathcal{C}}_0$, it is isomorphic to $L \circ u$ and therefore also admits a colimit in $\operatorname{\mathcal{C}}_0$ (Remark 7.1.1.8). $\square$

Warning 7.1.3.24. In the situation of Variant 7.1.3.23, the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ generally does not preserve the colimit of the diagram $u$. If $C = \varinjlim (u)$ is a colimit of $u$ in the $\infty$-category $\operatorname{\mathcal{C}}$, then $C$ usually does not belong to $\operatorname{\mathcal{C}}_0$. The colimit of $u$ in the $\infty$-category $\operatorname{\mathcal{C}}_0$ is instead given by the localization $L(C)$.

Corollary 7.1.3.25. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory of $\operatorname{\mathcal{C}}$, and let $K$ be a simplicial set. If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then $\operatorname{\mathcal{C}}_0$ also admits $K$-indexed limits. If $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then $\operatorname{\mathcal{C}}_0$ also admits $K$-indexed colimits.