Kerodon

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 4.6.4.19). 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. $\square$

Proof. Combine Remark 7.1.1.10 with Exercise 7.1.3.7. $\square$

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 $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. $\square$

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. $\square$

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 $(1)$. $\square$

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 $\operatorname{\mathcal{C}}_{/g}$. $\square$

Proof. Combine Propositions 7.1.3.19 and 7.1.3.18. $\square$

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. $\square$

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. $\square$

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). $\square$

Proof. Combine Corollary 7.1.3.22 with Variant 7.1.3.23. $\square$