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7.1.4 Relative Initial and Final Objects

In §4.6.6, we introduced the notions of initial and final object of an $\infty$-category $\operatorname{\mathcal{C}}$ (Definition 4.6.6.1). In this section, we study the more general notions of $U$-initial and $U$-final objects, where $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty$-categories.

Definition 7.1.4.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. We say that an object $Y \in \operatorname{\mathcal{C}}$ is $U$-final if, for every object $X \in \operatorname{\mathcal{C}}$, the functor $U$ induces a homotopy equivalence

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) ).$

We say that $Y$ is $U$-initial if, for every object $Z \in \operatorname{\mathcal{C}}$, the functor $U$ induces a homotopy equivalence

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), U(Z) ).$

Example 7.1.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $U: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ be the projection map. Then an object $Y \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if it is initial, and $U$-final if and only if it is final.

Remark 7.1.4.3. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $U$-initial objects. Then the restriction $U|_{\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is fully faithful. Similarly, $U$ is fully faithful when restricted to the full subcategory of $U$-final objects of $\operatorname{\mathcal{C}}$.

Example 7.1.4.4. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

• The functor $U$ is fully faithful.

• Every object of $\operatorname{\mathcal{C}}$ is $U$-initial.

• Every object of $\operatorname{\mathcal{C}}$ is $U$-final.

Remark 7.1.4.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories. Then an object $Y \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if it is $U^{\operatorname{op}}$-final, when regarded as an object of the opposite $\infty$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Remark 7.1.4.6 (Transitivity). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, and let $Y \in \operatorname{\mathcal{C}}$ be an object for which $U(Y)$ is $V$-final. Then $Y$ is $U$-final if and only if it is $(V \circ U)$-final.

Remark 7.1.4.7. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $Y \in \operatorname{\mathcal{C}}$ be an object. Suppose that $U(Y)$ is a final object of $\operatorname{\mathcal{D}}$. Then $Y$ is a final object of $\operatorname{\mathcal{C}}$ if and only if it is a $U$-final object of $\operatorname{\mathcal{C}}$ (apply Remark 7.1.4.6 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$).

Remark 7.1.4.8. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, and let $V: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be another functor which is isomorphic to $U$ (as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then an object $Y \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if it is $V$-initial. To prove this, let $Z$ be an object of $\operatorname{\mathcal{C}}$ and let $\alpha : U \rightarrow V$ be an isomorphism of functors, so that we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), U(Z) ) \ar [d]^{ [\alpha _ Z] \circ } \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( V(Y), V(Z) ) \ar [r]^-{ \circ [ \alpha _ Y ] } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), V(Z) ) }$

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the bottom horizontal and right vertical maps are homotopy equivalences. It follows that the upper horizontal map is a homotopy equivalence if and only if the left vertical map is a homotopy equivalence. Similarly, the object $Y$ is $U$-final if and only if it is $V$-final.

Remark 7.1.4.9. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{U} \ar [r]^-{F} & \operatorname{\mathcal{C}}' \ar [d]^{U'} \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}', }$

where the horizontal maps are equivalences of $\infty$-categories. Then an object $X \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if $F(X) \in \operatorname{\mathcal{C}}'$ is $U'$-initial, and $U$-final if and only if $F(X)$ is $U'$-final.

Proposition 7.1.4.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ with the property that $U(f)$ is an isomorphism. Then any two of the following three conditions imply the third:

$(1)$

The object $X$ is $U$-initial.

$(2)$

The object $Y$ is $U$-initial.

$(3)$

The morphism $f$ is an isomorphism.

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 4.6.8.15).

This follows from the commutativity of the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \ar [r]^-{ \circ [f] } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), U(Z) ) \ar [r]^-{ \circ [ U(f) ] } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Z) ) }$

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 7.1.4.10 follows by allowing the object $Z$ to vary. $\square$

Corollary 7.1.4.11. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, and let $f: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$. Then the object $X$ is $U$-initial if and only if $Y$ is $U$-initial, and the object $X$ is $U$-final if and only if $Y$ is $U$-final.

Corollary 7.1.4.12 (Uniqueness). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $X$ and $Y$ be $U$-initial objects of $\operatorname{\mathcal{C}}$. Then $X$ and $Y$ are isomorphic if and only if $U(X)$ and $U(Y)$ are isomorphic as objects of $\operatorname{\mathcal{D}}$.

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 7.1.4.10, we deduce that $f$ is an isomorphism. In particular, the objects $X$ and $Y$ are isomorphic. $\square$

Recall that a functor of $\infty$-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a coreflective localization if it admits a fully faithful left adjoint $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ (Proposition 6.3.3.13). This condition has a simple formulation in terms of relatively final objects:

Proposition 7.1.4.13. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then $U$ is a coreflective localization functor if and only if, 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 the $\infty$-category $\operatorname{\mathcal{D}}$.

We will deduce Proposition 7.1.4.13 from a slightly more precise result.

Lemma 7.1.4.14. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $\operatorname{\mathcal{C}}_0$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $F$-initial objects, and suppose that the restriction $U_0 = U|_{\operatorname{\mathcal{C}}_0}$ is essentially surjective. Then:

$(1)$

The functor $U_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty$-categories.

$(2)$

Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, where $X$ is $U$-initial. Then $e$ exhibits $X$ as a $\operatorname{\mathcal{C}}_0$-coreflection of $Y$ (in the sense of Definition 6.2.2.1) if and only if $U(e)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$.

$(3)$

The full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is coreflective.

$(4)$

Let $F_0: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_0$ be a homotopy inverse of the functor $U_0$, and let $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a composition of $F_0$ with the inclusion map $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$. Then $F$ is a left adjoint of $U$.

$(5)$

The functor $U$ is a coreflective localization.

Proof. Note that the functor $U_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is automatically fully faithful (Remark 7.1.4.3). 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

$\xymatrix { \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X) \ar [r]^-{\circ [e] } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(C), U(X) ) \ar [r]^-{ \circ [U(e)] } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(C), U(Y) ) }$

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 6.2.2.7, 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 $\operatorname{\mathcal{C}}_0$-colocalization 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 6.2.2.11). 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 6.3.3.13. $\square$

Proof of Proposition 7.1.4.13. 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 7.1.4.14). Using Proposition 6.3.3.13, 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

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(D), X ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (U \circ F)(D), U(X) ) \xrightarrow { \circ [\eta _ D] } \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, U(X) )$

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). $\square$

Corollary 7.1.4.15. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories. Then $U$ is a coreflective localization functor if and only if, for every object $Y \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{C}}_{Y} = \{ Y\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ contains an $U$-initial object of $\operatorname{\mathcal{C}}$.

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 7.1.4.13). Using Proposition 7.1.4.13, 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 7.1.4.11). $\square$

Proposition 7.1.4.16. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then:

$(1)$

An object $Y \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if $U$ induces an equivalence of $\infty$-categories $U': \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ U(Y) / }$.

$(2)$

An object $Y \in \operatorname{\mathcal{C}}$ is $U$-final if and only if $U$ induces an equivalence of $\infty$-categories $U'': \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/U(Y)}$.

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

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/Y} \ar [rr]^{U''} \ar [dr] & & \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / U(Y) } \ar [dl] \\ & \operatorname{\mathcal{C}}& }$

where the vertical maps are right fibrations (Proposition 4.3.6.1). Applying Corollary 5.1.6.15, 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

$U''_{X}: \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \{ X\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / U(Y) }$

is a homotopy equivalence of Kan complexes. By virtue of Proposition 4.6.5.9, this is equivalent to the requirement that $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) )$. $\square$

Corollary 7.1.4.17. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories and let $Y$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The object $Y$ is $U$-initial.

$(2)$

The induced map $U': \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ U(Y)/ }$ is a trivial Kan fibration.

$(3)$

Every lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}}$

has a solution, provided that $n > 0$ and $\sigma _0(0) = Y$.

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 7.1.4.16. The equivalence $(2) \Leftrightarrow (3)$ is immediate from the definitions. $\square$

Corollary 7.1.4.18. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory of $\operatorname{\mathcal{C}}$ whose objects are $U$-initial, and let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be the full subcategory of $\operatorname{\mathcal{D}}$ spanned by objects of the form $U(C)$ for $C \in \operatorname{\mathcal{C}}_0$. Then the functor $U|_{\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ is a trivial Kan fibration.

Proof. Suppose we are given a lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^{U_0} \\ \Delta ^ n \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur] & \operatorname{\mathcal{D}}_0. }$

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 7.1.4.17 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 $\operatorname{\partial \Delta }^{n}$). $\square$

Proposition 7.1.4.19. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr]_{U} & & \operatorname{\mathcal{D}}\ar [dl]^{V} \\ & \operatorname{\mathcal{E}}, & }$

where $U$ and $V$ are inner fibrations. Let $E \in \operatorname{\mathcal{E}}$ be an object, and let $F_{E}: \operatorname{\mathcal{C}}_{E} \rightarrow \operatorname{\mathcal{D}}_{E}$ denote the corresponding restriction of $F$. Then:

$(1)$

If $X \in \operatorname{\mathcal{C}}_{E}$ is $F$-initial when viewed as an object of the $\infty$-category $\operatorname{\mathcal{C}}$, then $X$ is $F_{E}$-initial.

$(2)$

Assume that $U$ and $V$ are cartesian fibrations, and that the functor $F$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{D}}$. If $X$ is $F_{E}$-initial, then it is $F$-initial when viewed as an object of $\operatorname{\mathcal{C}}$.

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

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \ar [rr]^{\rho } \ar [dr] & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \ar [dl] \\ & \operatorname{Hom}_{\operatorname{\mathcal{E}}}( E, E ). & }$

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 4.6.1.19). Applying Corollary 3.3.7.3, we conclude that $\rho$ restricts to a homotopy equivalence

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}_{E}}(X,Y) & = & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(E,E) } \{ \operatorname{id}_{E} \} \\ & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X),F(Y)) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(E,E) } \{ \operatorname{id}_{E} \} \\ & = & \operatorname{Hom}_{\operatorname{\mathcal{D}}_{E}}( F(X), F(Y) ). \end{eqnarray*}

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

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Z ) \ar [rr]^{\theta } \ar [dr] & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) \ar [dl] \\ & \operatorname{Hom}_{\operatorname{\mathcal{E}}}( U(X), U(Z) ) & }$

is a homotopy equivalence. Since the vertical maps are Kan fibrations (Proposition 4.6.1.19), it will suffice to show that the induced map

$\theta _{ \overline{f}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( U(X), U(Z) ) } \{ \overline{f} \} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( U(X), U(Z) ) } \{ \overline{f} \}$

is a homotopy equivalence, for each morphism $\overline{f}: U(X) \rightarrow U(Z)$ in the $\infty$-category $\operatorname{\mathcal{E}}$ (Corollary 3.3.7.3). 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 5.1.2.1, we can replace $\theta _{ \overline{f} }$ with the morphism

$\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{E} }( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}_{E} }( F(X), F(Y) ),$

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

Exercise 7.1.4.20. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty$-categories, and let $C \in \operatorname{\mathcal{C}}$ be an object having image $D = U(C)$ in $\operatorname{\mathcal{D}}$. Show that $C$ is $U$-initial if and only if the following condition is satisfied:

$(\ast )$

For every morphism $f: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the covariant transport functor $f_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ carries $C$ to an initial object of the $\infty$-category $\operatorname{\mathcal{C}}_{D'}$.

For a more general statement, see Proposition 7.3.8.2.

Corollary 7.1.4.21. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories, and let $C \in \operatorname{\mathcal{C}}$ be an object having image $D = U(C)$ in $\operatorname{\mathcal{D}}$. If the object $C$ is $U$-initial, then it is initial when regarded as an object of the $\infty$-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. The converse holds if $U$ is a cartesian fibration.

Proof. Apply Proposition 7.1.4.19 in the special case where $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}' = \{ D\}$. $\square$

Corollary 7.1.4.22. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of $\infty$-categories. The following conditions are equivalent:

$(1)$

For each object $D \in \operatorname{\mathcal{D}}$, the $\infty$-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ has an initial object.

$(2)$

The functor $U$ is a coreflective localization: that is, it admits a fully faithful left adjoint $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.