Kerodon

Proof. Fix an object $Z \in \operatorname{\mathcal{C}}$. We claim that any two of the following three conditions imply the third:

$(1_{Z})$

The functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) )$.

$(2_{Z})$

The functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(Y), U(Z) )$.

$(3_{Z})$

Precomposition $[f]$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ (see Notation 4.6.9.15).

This follows from the commutativity of the diagram

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, since the bottom horizontal map is a homotopy equivalence (by virtue of our assumption that $U(f)$ is an isomorphism). Proposition 7.1.5.10 follows by allowing the object $Z$ to vary. $\square$

Proof. Assume that there exists an isomorphism $\overline{f}: U(X) \rightarrow U(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$. Since $X$ is $U$-initial, the functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) )$. It follows that there exists a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ such that $U(f)$ is homotopic to $\overline{f}$. In particular, $U(f): U(X) \rightarrow U(Y)$ is also an isomorphism in $\operatorname{\mathcal{D}}$. Applying Proposition 7.1.5.10, we deduce that $f$ is an isomorphism. In particular, the objects $X$ and $Y$ are isomorphic. $\square$

Proof. Note that the functor $U_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is automatically fully faithful (Remark 7.1.5.3). Our assumption that $U_0$ is essentially surjective then guarantees that it is an equivalence of $\infty $-categories, which proves $(1)$.

We next prove the following:

$(\ast )$

For every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $e: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, where $X$ is $U$-initial and $U(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.

To prove $(\ast )$, we observe that the essential surjectivity of $U_0$ guarantees that there exists a $U$-initial object $X \in \operatorname{\mathcal{C}}$ and an isomorphism $\overline{e}: U(X) \rightarrow U(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$. Since $X$ is $U$-initial, the functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) )$. Modifying $\overline{e}$ by a homotopy, we can assume without loss of generality that $\overline{e} = U(e)$ for some morphism $X \rightarrow Y$ of $\operatorname{\mathcal{C}}$.

We now prove $(2)$. Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, where the object $X$ is $U$-initial. Assume first that $U(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$. We wish to show that, for every $U$-initial object $C \in \operatorname{\mathcal{C}}$, postcomposition with $e$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$. This follows by inspecting the commutative diagram

in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$; here the vertical maps are homotopy equivalences by virtue of our assumption that $C$ is $U$-initial, and the bottom horizontal map is a homotopy equivalence by virtue of our assumption that $U(e)$ is an isomorphism.

We now prove the converse. Assume that $e: X \rightarrow Y$ exhibits $X$ as a $\operatorname{\mathcal{C}}_0$-coreflection of $Y$; we wish to show that $U(e)$ is an isomorphism. Using $(\ast )$, we can choose a $U$-initial object $X' \in \operatorname{\mathcal{C}}$ and a morphism $e': X' \rightarrow Y$ such that $U(e')$ is an isomorphism in $\operatorname{\mathcal{D}}$. It follows from the previous step that $e'$ exhibits $X'$ as a $\operatorname{\mathcal{C}}_0$-coreflection of $Y$. It follows that $e$ can be realized as the composition of $e'$ with an isomorphism $v: X \rightarrow X'$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Remark 6.2.2.3). Then $U(e)$ is a composition of the isomorphisms $U(v)$ and $U(e')$ in the $\infty $-category $\operatorname{\mathcal{D}}$, and is therefore also an isomorphism.

Assertion $(3)$ follows immediately from $(2)$ and $(\ast )$. Combining $(3)$ with Proposition 6.2.2.13, we see that there exists a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ and a natural transformation $\eta : L \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ which exhibits $L$ as a $\operatorname{\mathcal{C}}_0$-coreflection functor: that is, it carries each object $Y \in \operatorname{\mathcal{C}}$ to a morphism $\eta _{Y}: L(Y) \rightarrow Y$ where $L(Y)$ is $U$-initial and $U( \eta _ Y)$ is an isomorphism. In particular, $\eta $ induces an isomorphism $U_0 \circ L \rightarrow U$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (Theorem 4.4.4.4). It follows from assumption $(1)$ that the functor $U_0$ admits a homotopy inverse $F_0: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_0$, which is also a left adjoint of $U_0$ (Example 6.2.1.11). Moreover, the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is left adjoint to $L$ (Proposition 6.2.2.17). It follows that the composition $F = \iota \circ F_0$ is left adjoint to $U_0 \circ L$ (Remark 6.2.1.8), and therefore also to $U$. This proves $(4)$. Moreover, the functor $F$ is fully faithful (since $F_0$ is an equivalence of $\infty $-categories and $\iota $ is the inclusion of a full subcategory), so assertion $(5)$ follows from Proposition 6.3.3.6. $\square$

Proof of Proposition 7.1.5.13. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $U$ is a coreflective localization functor: we will show that, for every object $D \in \operatorname{\mathcal{D}}$, there exists a $U$-initial object $C \in \operatorname{\mathcal{C}}$ and an isomorphism $D \rightarrow U(C)$ in $\operatorname{\mathcal{D}}$ (the converse follows from Lemma 7.1.5.14). Using Proposition 6.3.3.6, we see that there exists a functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural isomorphism $\eta : \operatorname{id}_{\operatorname{\mathcal{D}}} \rightarrow U \circ F$ which is the unit of an adjunction between $F$ and $U$. In particular, for every object $D \in \operatorname{\mathcal{D}}$, we have an isomorphism $\eta _{D}: D \rightarrow U(C)$ for $C = F(D)$. We will complete the proof by showing that the object $C$ is $U$-initial. Fix an object $X \in \operatorname{\mathcal{C}}$; we wish to show that the functor $U$ induces a homotopy equivalence of Kan complexes $\rho : \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(D), X ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (U \circ F)(D), U(X) )$. Since $\eta _{D}: D \rightarrow (U \circ F)(D)$ is an isomorphism, this is equivalent to the requirement that the composite map

is a homotopy equivalence of Kan complexes, which follows from our assumption that $\eta $ is the unit of an adjunction (Proposition 6.2.1.17). $\square$

Proof. Assume that $U$ is a coreflective localization functor. We will show that, for each object $Y \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{Y}$ contains a $U$-initial object of $\operatorname{\mathcal{C}}$ (the converse follows immediately from Proposition 7.1.5.13). Using Proposition 7.1.5.13, we see that there exists a $U$-initial object $X \in \operatorname{\mathcal{C}}$ and an isomorphism $e: Y \rightarrow U(X)$ in $\operatorname{\mathcal{D}}$. Since $U$ is an isofibration, we can lift $e$ to an isomorphism $\widetilde{e}: \widetilde{Y} \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $X$ is $U$-initial then guarantees that $\widetilde{Y}$ is also $U$-initial (Corollary 7.1.5.11). $\square$

Proof. We will prove $(2)$; the proof of $(1)$ is similar. Fix an object $Y \in \operatorname{\mathcal{C}}$, so that the morphism $U''$ of $(1)$ fits into a commutative diagram

where the vertical maps are right fibrations (Proposition 4.3.6.1). Applying Corollary 5.1.7.16, we see that $U''$ is an equivalence of $\infty $-categories if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the induced map of fibers

is a homotopy equivalence of Kan complexes. By virtue of Proposition 4.6.5.10, this is equivalent to the requirement that $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) )$. $\square$

Proof. Since $U$ is an inner fibration, the morphism $U'$ is a left fibration (Corollary 4.3.6.9). In particular, it is a trivial Kan fibration if and only if it is an equivalence of $\infty $-categories (Proposition 4.5.5.20). The equivalence $(1) \Leftrightarrow (2)$ now follows from Proposition 7.1.5.16. The equivalence $(2) \Leftrightarrow (3)$ is immediate from the definitions. $\square$

Proof. Suppose we are given a lifting problem

If $n=0$, this lifting problem admits a solution by the definition of the subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$. If $n > 0$, then $\sigma _0(0)$ is a $U$-initial object of $\operatorname{\mathcal{C}}$, so Corollary 7.1.5.17 guarantees that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $U(\sigma ) = \overline{\sigma }$. We conclude by observing that $\sigma $ automatically factors through the full subcategory $\operatorname{\mathcal{C}}_0$ (since every vertex of $\Delta ^ n$ is contained in the boundary $\operatorname{\partial \Delta }^{n}$). $\square$

Proof. We first prove $(1)$. Assume that $X$ is $F$-initial. For every object $Y \in \operatorname{\mathcal{C}}_{E}$, we have a commutative diagram of Kan complexes

Our assumption that $X$ is $F$-initial guarantees that $\rho $ is a homotopy equivalence. Since $U$ and $V$ are inner fibrations, the vertical maps are Kan fibrations (Proposition 4.6.1.21). Applying Corollary 3.3.7.5, we conclude that $\rho $ restricts to a homotopy equivalence

Allowing $Y$ to vary over objects of $\operatorname{\mathcal{C}}_{E}$, it follows that $X$ is an $F_{E}$-initial object of $\operatorname{\mathcal{C}}$.

We now prove $(2)$. Assume that $U$ and $V$ are cartesian fibrations, that the functor $F$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{D}}$, and that $X$ is $F_{E}$-initial. We wish to show that $X$ is $F$-initial. Fix an object $Z \in \operatorname{\mathcal{C}}$; we must show that the horizontal map in the diagram

is a homotopy equivalence. Since the vertical maps are Kan fibrations (Proposition 4.6.1.21), it will suffice to show that the induced map

is a homotopy equivalence, for each morphism $\overline{f}: U(X) \rightarrow U(Z)$ in the $\infty $-category $\operatorname{\mathcal{E}}$ (Corollary 3.3.7.5). Since $U$ is a cartesian fibration, we can write $\overline{f} = U(f)$, where $f: Y \rightarrow Z$ is a $U$-cartesian morphism in $\operatorname{\mathcal{C}}$. By assumption, the image $F(f): F(Y) \rightarrow F(Z)$ is a $V$-cartesian morphism in the $\infty $-category $\operatorname{\mathcal{D}}$. Using Proposition 5.1.3.11, we can replace $\theta _{ \overline{f} }$ with the morphism

which is a homotopy equivalence by virtue of our assumption that $X$ is $F_{E}$-initial. $\square$

Proof. Apply Proposition 7.1.5.19 in the special case where $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}' = \{ D\} $. $\square$

Proof. Combine Corollaries 7.1.5.15 and 7.1.5.21. $\square$