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7.1.6 Relative Initial and Final Objects

In ยง7.1.2, we introduced the notions of initial and final object of an $\infty $-category $\operatorname{\mathcal{C}}$ (Definition 7.1.2.1). 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 7.1.6.1. 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 7.1.6.2. 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.

Example 7.1.6.3. 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 7.1.6.4. 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 7.1.6.5 (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 7.1.6.6. 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 7.1.6.5 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$).

Remark 7.1.6.7. 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 7.1.6.8. 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 7.1.6.9. 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 4.6.7.15).

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

Corollary 7.1.6.10. 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 $U$ is $U$-final.

Corollary 7.1.6.11 (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 7.1.6.9, we deduce that $f$ is an isomorphism. In particular, the objects $X$ and $Y$ are isomorphic. $\square$

Proposition 7.1.6.12. 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 7.1.6.13. 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 4.5.7.16). The equivalence $(1) \Leftrightarrow (2)$ now follows from Proposition 7.1.6.12. The equivalence $(2) \Leftrightarrow (3)$ is immediate from the definitions. $\square$

Corollary 7.1.6.14. 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 essential image of the functor $U|_{\operatorname{\mathcal{C}}_0}$. 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 7.1.6.13 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 7.1.6.15. 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. $\square$

Corollary 7.1.6.16. 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 7.1.6.15 in the special case where $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}' = \{ D\} $. $\square$

Exercise 7.1.6.17. 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 7.4.8.2.