Definition 7.1.4.1. 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 $\operatorname{\mathcal{D}}$.
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:
Remark 7.1.4.2. In the situation of Definition 7.1.4.1, 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 7.1.3.14).
Remark 7.1.4.3. 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 7.1.4.4. 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 7.1.4.5. 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 7.1.3.10, this is equivalent to the observation that $F$ carries isomorphisms in $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{D}}$ (see Remark 1.5.1.6).
Warning 7.1.4.6. In the formulation of Definition 7.1.4.4, 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 7.1.4.4 in cases where the $\infty $-category admits $K$-indexed limits or colimits, so that the conclusion of Definition 7.1.4.4 is non-vacuous.
Exercise 7.1.4.7. 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 $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$.
Remark 7.1.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let be a diagram which admits a limit $C = \varprojlim _{k \in K} C_ k$. Suppose we are given a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ such that the diagram $(F \circ q): K \rightarrow \operatorname{\mathcal{D}}$ admits a limit $D = \varprojlim _{k \in K} F(C_ k)$. Choose natural transformations which exhibit $C$ as a limit of $q$ and $D$ as a limit $F \circ q$, respectively. Invoking the universal property of $\beta $, we see that there is a morphism $\gamma : F(C) \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}$ and a commutative diagram in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{D}})$; moreover, $\gamma $ is unique up to homotopy. Stated more informally, there is a natural comparison map which is an isomorphism if and only if the functor $F$ preserves the limit of $q$.
Variant 7.1.4.9. It will often be useful to extend the terminology of Definition 7.1.4.4, 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 7.1.4.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $K$ be a simplicial set. Then:
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 $\operatorname{\mathcal{D}}$.
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
Since $F$ is an equivalence of $\infty $-categories, the horizontal maps in this diagram are also equivalences of $\infty $-categories (Corollary 4.6.4.21). 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 7.1.3.12. $\square$
Variant 7.1.4.11. 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 $\operatorname{\mathcal{C}}$.
Proof. Combine Remark 7.1.1.10 with Exercise 7.1.4.7. $\square$
Corollary 7.1.4.12. 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:
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 $K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$.
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 7.1.4.10 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 7.1.4.10), 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 7.1.3.15, we deduce that $u$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. $\square$
Corollary 7.1.4.13. 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 7.1.4.14. 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:
The composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram.
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 4.6.7.15).
A conservative functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which preserves $K$-indexed limits also reflects them:
Proposition 7.1.4.15. 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 7.1.4.15 is an immediate consequence of the following more precise assertion:
Lemma 7.1.4.16. 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 $\operatorname{\mathcal{D}}_{ / (F \circ u)}$. In fact, the functor $F_{/u}$ is conservative: this follows from our assumption that $F$ is conservative, by virtue of Corollary 4.4.2.12. $\square$
Definition 7.1.4.17. 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 7.1.4.18. 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 7.1.4.16).
Proposition 7.1.4.19. 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:
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits and the functor $F$ preserves $K$-indexed limits.
The functor $F$ creates $K$-indexed limits.
Proof. The implication $(1) \Rightarrow (2)$ is immediate. 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 7.1.4.14, 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 7.1.4.17:
Proposition 7.1.4.20. 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:
The projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ creates $K$-indexed limits, for every simplicial set $K$.
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 7.1.3.12), 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 4.6.7.12, since $\overline{g}$ is a final object of $\operatorname{\mathcal{C}}_{/g}$. $\square$
Corollary 7.1.4.21. 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:
If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ admits $K$-indexed limits. Moreover, a morphism $K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{f/}$ is a limit diagram if and only if its image in $\operatorname{\mathcal{C}}$ is a limit diagram.
If $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ admits $K$-indexed colimits. Moreover, a morphism $K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a colimit diagram if and only if its image in $\operatorname{\mathcal{C}}$ is a colimit diagram.
Proof. Combine Propositions 7.1.4.20 and 7.1.4.19 with Remark 7.1.4.18. $\square$
Corollary 7.1.4.22. 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 6.2.6.6 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 7.1.4.23 (Colimits in a Reflective Localization). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory (Definition 6.2.2.6), and let $u: K \rightarrow \operatorname{\mathcal{C}}'$ be a diagram. If $u$ admits a colimit in $\operatorname{\mathcal{C}}$, then it also admits a colimit in $\operatorname{\mathcal{C}}'$.
Proof. By virtue of Proposition 6.2.2.18, the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$. If $u$ admits a colimit in $\operatorname{\mathcal{C}}$, then $L \circ u$ admits a colimit in $\operatorname{\mathcal{C}}'$ (Corollary 7.1.4.22). Since $u$ factors through $\operatorname{\mathcal{C}}'$, it is isomorphic to $L \circ u$ and therefore also admits a colimit in $\operatorname{\mathcal{C}}'$ (Remark 7.1.1.8). $\square$
Warning 7.1.4.24. In the situation of Corollary 7.1.4.23, the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ 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 $\operatorname{\mathcal{C}}'$. The colimit of $u$ in the $\infty $-category $\operatorname{\mathcal{C}}'$ is instead given by the localization $L(C)$.
Variant 7.1.4.25 (Limits in a Reflective Localization). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory. Then a diagram $u: K \rightarrow \operatorname{\mathcal{C}}'$ admits a limit in $\operatorname{\mathcal{C}}'$ 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 $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$.
We will deduce Variant 7.1.4.25 from the following special case:
Lemma 7.1.4.26. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory. If $\operatorname{\mathcal{C}}$ contains a final object $X$, then $\operatorname{\mathcal{C}}'$ contains an object which is isomorphic to $X$. In particular, if $\operatorname{\mathcal{C}}'$ is replete, then it contains every final object of $\operatorname{\mathcal{C}}$.
Proof. Choose a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ which exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ (see Definition 6.2.2.6). Since $X$ is a final object of $\operatorname{\mathcal{C}}$, we can choose a morphism $g: Y \rightarrow X$. We will complete the proof by showing that $g$ is a homotopy inverse to $f$: that is, the homotopy classes $[f]$ and $[g]$ are inverses of one another in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Since $X$ is a final object of $\operatorname{\mathcal{C}}$, the equality $[g] \circ [f] = [ \operatorname{id}_ X ]$ is automatic. We wish to prove the equality $[f] \circ [g] = [ \operatorname{id}_ Y ]$. Since $f$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, precomposition with $[f]$ induces a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, Y ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Y)$. The desired result now follows from the calculation
Proof of Variant 7.1.4.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $u: K \rightarrow \operatorname{\mathcal{C}}'$ be a diagram. Applying Corollary 6.2.2.17, we deduce that $\operatorname{\mathcal{C}}'_{/u}$ is a reflective subcategory of $\operatorname{\mathcal{C}}_{/u}$. If the diagram $u$ admits a limit in $\operatorname{\mathcal{C}}$, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$ has a final object $X$. Applying Lemma 7.1.4.26, we deduce that $X$ is isomorphic to an object of $\operatorname{\mathcal{C}}'_{/u}$, which is also a final object of $\operatorname{\mathcal{C}}_{/u}$ and therefore also of $\operatorname{\mathcal{C}}'_{/u}$. In particular, the diagram $u$ has a limit in $\operatorname{\mathcal{C}}'$ (which is preserved by the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$). $\square$
Corollary 7.1.4.27. 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.
Proof. Combine Variant 7.1.4.25 with Corollary 7.1.4.23. $\square$