Definition 6.2.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. We say that a morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is a $\operatorname{\mathcal{C}}'$-local equivalence if, for every object $Z \in \operatorname{\mathcal{C}}'$, precomposition with $u$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [u] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
6.2.2 Reflective Subcategories
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Our goal in this section is to characterize those full subcategories $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ for which the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left or right adjoint. We begin by introducing some terminology.
Warning 6.2.2.2. Definition 6.2.2.1 is not self-dual. We say that a morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is a $\operatorname{\mathcal{C}}'$-colocal equivalence if, for every object $W \in \operatorname{\mathcal{C}}'$, composition with $u$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \xrightarrow { [u] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)$. In other words, $u$ is a $\operatorname{\mathcal{C}}'$-colocal equivalence if it is a $\operatorname{\mathcal{C}}'^{\operatorname{op}}$-local equivalence when viewed as a morphism in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Beware that this is usually not equivalent to the requirement that $u$ is a $\operatorname{\mathcal{C}}'$-local equivalence (see Example 6.2.2.3).
Example 6.2.2.3. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1) and let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between small $\infty $-categories, which we regard as a morphism in the $\infty $-category $\operatorname{\mathcal{QC}}$. Then:
The functor $F$ is a $\operatorname{\mathcal{S}}$-local equivalence (in the sense of Definition 6.2.2.1) if and only if it is a weak homotopy equivalence of simplicial sets.
The functor $F$ is a $\operatorname{\mathcal{S}}$-colocal equivalence (in the sense of Warning 6.2.2.2) if and only if the map of cores $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a homotopy equivalence.
See Remark 5.5.4.6 and Proposition 4.4.3.17.
Remark 6.2.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If $u$ is an isomorphism, then it is a $\operatorname{\mathcal{C}}'$-local equivalence. Conversely, if $X$ and $Y$ belong to $\operatorname{\mathcal{C}}'$ and $u$ is a $\operatorname{\mathcal{C}}'$-local equivalence, then it is an isomorphism.
Remark 6.2.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a commutative diagram and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If any two of the morphisms $u$, $v$, and $w$ are $\operatorname{\mathcal{C}}'$-local equivalences, then so is the third.
\[ \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]
Definition 6.2.2.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. We say that a morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ if $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}'$ and $u$ is a $\operatorname{\mathcal{C}}'$-local equivalence. We say that $u$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$ if $X$ belongs to $\operatorname{\mathcal{C}}'$ and $u$ is a $\operatorname{\mathcal{C}}'$-colocal equivalence. We say that a subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective if it is full and, for every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $u: X \rightarrow Y$ which exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. We say that the subcategory $\operatorname{\mathcal{C}}'$ is coreflective if it is full and, for every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $u: X \rightarrow Y$ which exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$.
Remark 6.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, so that we can identify $\operatorname{\mathcal{C}}'^{\operatorname{op}}$ with a full subcategory of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then:
A morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ if and only if $u^{\operatorname{op}}: Y^{\operatorname{op}} \rightarrow X^{\operatorname{op}}$ exhibits $Y^{\operatorname{op}}$ as a $\operatorname{\mathcal{C}}'^{\operatorname{op}}$-coreflection of $X^{\operatorname{op}}$.
The subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective if and only if the subcategory $\operatorname{\mathcal{C}}'^{\operatorname{op}} \subseteq \operatorname{\mathcal{C}}^{\operatorname{op}}$ is coreflective.
Remark 6.2.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and suppose we are given a pair of morphisms $u: X \rightarrow Y$ and $w: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$, where $Y$ and $Z$ belong to the subcategory $\operatorname{\mathcal{C}}'$. If $u$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, then we can realize $w$ as a composition of $u$ with another morphism $v: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}'$, which is uniquely determined up to homotopy. Moreover, $v$ is an isomorphism if and only if $w$ exhibits $Z$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ (see Remarks 6.2.2.5 and 6.2.2.4). Stated more informally: a $\operatorname{\mathcal{C}}'$-reflection of $X$, if it exists, is unique up to isomorphism.
Example 6.2.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If $X$ belongs to the subcategory $\operatorname{\mathcal{C}}'$, then $u$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ if and only if it is an isomorphism. Similarly, if $Y$ belongs to $\operatorname{\mathcal{C}}'$, then $u$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$ if and only if it is an isomorphism.
Example 6.2.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which contains a final object, and let $\operatorname{\mathcal{C}}^{\mathrm{fin}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its final objects (so that $\operatorname{\mathcal{C}}^{\mathrm{fin}}$ is a contractible Kan complex: see Corollary 4.6.7.14). Then $\operatorname{\mathcal{C}}^{\mathrm{fin}}$ is a reflective subcategory of $\operatorname{\mathcal{C}}$. Moreover, every morphism in $\operatorname{\mathcal{C}}$ is a $\operatorname{\mathcal{C}}^{\mathrm{fin}}$-local equivalence.
Example 6.2.2.11. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1) and let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). Then $\operatorname{\mathcal{S}}$ is a reflective and coreflective subcategory of $\operatorname{\mathcal{QC}}$. If $\operatorname{\mathcal{C}}$ is a small $\infty $-category, then the inclusion map $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ exhibits the core $\operatorname{\mathcal{C}}^{\simeq }$ as a $\operatorname{\mathcal{S}}$-coreflection of $\operatorname{\mathcal{C}}$ (see Example 6.2.2.3), and the comparision map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Ex}^{\infty }(\operatorname{\mathcal{C}})$ exhibits the Kan complex $\operatorname{Ex}^{\infty }(\operatorname{\mathcal{C}})$ as a $\operatorname{\mathcal{S}}$-reflection of $\operatorname{\mathcal{C}}$ (see Proposition 3.3.6.7).
Example 6.2.2.12. Let $f: X \rightarrow Y$ be a map of Kan complexes, which we regard as a morphism in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$. For every integer $n$, the following conditions are equivalent:
The morphism $f$ exhibits $Y$ as an $n$-truncation of $X$: that is, $Y$ is $n$-truncated and $f$ is $(n+1)$-connective (see Definition 3.5.7.19)
The morphism $f$ exhibits $Y$ as a $\operatorname{\mathcal{S}}_{\leq n}$-reflection of $X$, where $\operatorname{\mathcal{S}}_{\leq n} \subset \operatorname{\mathcal{S}}$ is the full subcategory spanned by the $n$-truncated Kan complexes.
This follows from the characterization of $n$-truncations supplied by Proposition 3.5.7.29 (together with Remark 5.5.1.5). For example, if $n \geq 0$, we can regard the fundamental $n$-groupoid $\pi _{\leq n}(X)$ as a $\operatorname{\mathcal{S}}_{\leq n}$-reflection of $X$ (Example 3.5.7.25). It follows that $\operatorname{\mathcal{S}}_{\leq n}$ is a reflective subcategory of $\operatorname{\mathcal{C}}$.
Variant 6.2.2.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between (small) $\infty $-categories, which we regard as a morphism in the $\infty $-category $\operatorname{\mathcal{QC}}$. For every integer $n \geq -1$, the following conditions are equivalent:
The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $(n-1)$-truncation of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.2.10.
The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a $\operatorname{\mathcal{QC}}_{\leq n}$-reflection of $\operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{QC}}_{\leq n} \subset \operatorname{\mathcal{QC}}$ denotes the full subcategory spanned by those $\infty $-categories which are locally $(n-1)$-truncated.
This follows from characterization of local $(n-1)$-truncations supplied by Proposition 4.8.2.19 (together with Remark 5.5.4.5). For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then we can regard the homotopy $n$-category $\mathrm{h}_{\mathit{\leq {}n}}\mathit{(\operatorname{\mathcal{C}})}$ as a $\operatorname{\mathcal{QC}}_{\leq n}$-reflection of $\operatorname{\mathcal{C}}$ (Corollary 4.8.4.8). It follows that $\operatorname{\mathcal{QC}}_{\leq n}$ is a reflective subcategory of $\operatorname{\mathcal{QC}}$.
Example 6.2.2.14. Let $\operatorname{Top}$ denote the category whose objects are topological spaces and whose morphisms are continuous functions. Let us regard $\operatorname{Top}$ as a simplicial category (Example 2.4.1.5), and let $\operatorname{\mathcal{T}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$ denote its homotopy coherent nerve. Let $\operatorname{\mathcal{T}}_{0} \subseteq \operatorname{\mathcal{T}}$ be the full subcategory spanned by those topological spaces which have the homotopy type of a CW complex. Then:
A continuous function between topological spaces $f: X \rightarrow Y$ is a weak homotopy equivalence (in the sense of Definition 3.6.3.1) if and only if it exhibits $X$ as a $\operatorname{\mathcal{T}}_0$-colocalization of $Y$. This is restatement of Corollary 3.6.5.4.
The full subcategory $\operatorname{\mathcal{T}}_{0} \subseteq \operatorname{\mathcal{T}}$ is coreflective. That is, for every topological space $Y$, there exists a weak homotopy equivalence $f: X \rightarrow Y$, where $X$ has the homotopy type of a CW complex. For example, we can take $f$ to be the counit map $| \operatorname{Sing}_{\bullet }(Y) | \rightarrow Y$ (see Corollary 3.6.4.2).
Definition 6.2.2.1 can be rephrased as a lifting property:
Proposition 6.2.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The morphism $f$ is a $\operatorname{\mathcal{C}}'$-local equivalence, in the sense of Definition 6.2.2.1.
For every object $Z \in \operatorname{\mathcal{C}}'$, the restriction map $\operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $ is a homotopy equivalence of Kan complexes.
The restriction map $u: \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}'$ is an equivalence of $\infty $-categories.
The restriction map $u$ is a trivial Kan fibration.
For $n \geq 2$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$, provided that $\sigma _0$ carries the initial edge $\Delta ^1 = \operatorname{N}_{\bullet }( \{ 0 < 1 \} )$ to the morphism $f$ and satisfies $\sigma _0(i) \in \operatorname{\mathcal{C}}'$ for $i \geq 2$.
Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.6.9.16, the equivalence $(2) \Leftrightarrow (3)$ from Corollary 5.1.7.16 (and Proposition 5.1.7.5). Corollary 4.3.6.12 guarantees that $u$ is a left fibration. In particular, it is an isofibration, so the equivalence $(3) \Leftrightarrow (4)$ is a special case of Proposition 4.5.5.20. The equivalence $(4) \Leftrightarrow (5)$ follows by unwinding definitions. $\square$
Corollary 6.2.2.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $q: K \rightarrow \operatorname{\mathcal{C}}'$ be a morphism of simplicial sets. Let $\overline{f}: \overline{X} \rightarrow \overline{Y}$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}_{/q}$, having image $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$. If $f$ is a $\operatorname{\mathcal{C}}'$-local equivalence in $\operatorname{\mathcal{C}}$, then then $\overline{f}$ is a $\operatorname{\mathcal{C}}'_{/q}$-local equivalence in $\operatorname{\mathcal{C}}_{/q}$.
Proof. Set $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{f/} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}'$. The assumption that $f$ is a $\operatorname{\mathcal{C}}'$-local equivalence guarantees that the restriction map $u: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a trivial Kan fibration (Proposition 6.2.2.15). We wish to show that the analogous restriction map
\[ \overline{u}: ( \operatorname{\mathcal{C}}_{/q} )_{ \overline{f} / } \times _{ \operatorname{\mathcal{C}}_{/q} } \operatorname{\mathcal{C}}'_{/q} \rightarrow ( \operatorname{\mathcal{C}}_{/q} )_{ \overline{X} / } \times _{ \operatorname{\mathcal{C}}_{/q} } \operatorname{\mathcal{C}}'_{/q} \]
is also trivial Kan fibration. Let us regard $\overline{f}$ as a morphism of simplicial sets $\Delta ^1 \star K \rightarrow \operatorname{\mathcal{C}}$, which we can identify with a diagram $\overline{q}: K \rightarrow \operatorname{\mathcal{D}}$. Under this identification, $\overline{u}$ corresponds to the map $\operatorname{\mathcal{D}}_{ / \overline{q} } \rightarrow \operatorname{\mathcal{E}}_{ / u \circ \overline{q} }$ induced by $u$, which is a trivial Kan fibration by virtue of Corollary 4.3.7.17. $\square$
Corollary 6.2.2.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $q: K \rightarrow \operatorname{\mathcal{C}}'$ be a diagram. If $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$, then $\operatorname{\mathcal{C}}'_{/q}$ is a reflective subcategory of $\operatorname{\mathcal{C}}_{/q}$.
Proof. Let $\overline{X}$ be an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/q}$; we wish to show that there exists a morphism $\overline{f}: \overline{X} \rightarrow \overline{Y}$ which exhibits $\overline{Y}$ as a $\operatorname{\mathcal{C}}'_{/q}$-reflection of $\overline{X}$. Let $X$ denote the image of $\overline{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$, we can choose a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ which exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. By virtue of Corollary 6.2.2.16, it will suffice to show that $f$ can be lifted to a morphism $\overline{f}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}_{/q}$. Unwinding the definitions, we can rewrite this as a lifting problem
\[ \xymatrix@C =50pt@R=50pt{ \emptyset \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}' \ar [d] \\ K \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}', } \]
which admits a solution by virtue of the fact that the right vertical map is a trivial Kan fibration (Proposition 6.2.2.15). $\square$
Our next goal is to prove the following:
Proposition 6.2.2.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion map. Then $\iota $ admits a left adjoint if and only if $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$. Similarly, $\iota $ admits a right adjoint if and only if $\operatorname{\mathcal{C}}'$ is a coreflective subcategory of $\operatorname{\mathcal{C}}$.
The first step toward proving Proposition 6.2.2.18 is to show that if $X \in \operatorname{\mathcal{C}}$ is an object which admits a $\operatorname{\mathcal{C}}'$-reflection $u: X \rightarrow Y$, then the pair $(u,Y)$ can be chosen to depend functorially on $X$.
Definition 6.2.2.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be a functor. We will say that a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor if, for every object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _{X}: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $\operatorname{\mathcal{C}}$, in the sense of Definition 6.2.2.6. We say that a natural transformation $\epsilon : L \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-coreflection functor if, for every object $Y \in \operatorname{\mathcal{C}}$, the morphism $\epsilon _{Y}: L(Y) \rightarrow Y$ exhibits $L(Y)$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$.
Remark 6.2.2.20. In the situation of Definition 6.2.2.19, the assumption that $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor guarantees in particular that for every object $X \in \operatorname{\mathcal{C}}$, the image $L(X)$ belongs to the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. Consequently, we can also view $L$ as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{C}}'$.
Remark 6.2.2.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be a $\operatorname{\mathcal{C}}'$-reflection functor. Then a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is a $\operatorname{\mathcal{C}}'$-local equivalence (in the sense of Definition 6.2.2.1) if and only if $L(f)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}'$.
Lemma 6.2.2.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $\operatorname{\mathcal{C}}'$ is reflective if and only if there exists a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor.
Proof. Assume that $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$; we will show that there exists a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor (the reverse implication is immediate from the definitions). Let $\operatorname{\mathcal{E}}$ be the full subcategory of $\operatorname{\mathcal{C}}\times \Delta ^1$ spanned by those objects $(X,i)$ having the property that if $i=1$, then $X$ belongs to the full subcategory $\operatorname{\mathcal{C}}'$. Let $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ denote the projection map. Let $\widetilde{u}: (X,0) \rightarrow (Y,1)$ be a morphism in $\operatorname{\mathcal{E}}$, corresponding to a morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ for which the target $Y$ belongs to $\operatorname{\mathcal{C}}'$. By virtue of Corollary 5.1.2.3, the morphism $\widetilde{u}$ is $\pi $-cocartesian if and only if $u$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. Consequently, our assumption that $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$ guarantees that $\pi $ is a cocartesian fibration of $\infty $-categories. Applying Proposition 5.2.2.8, we deduce that there exists a functor
\[ L: \operatorname{\mathcal{C}}\simeq \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}\rightarrow \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}\simeq \operatorname{\mathcal{C}}' \]
and a morphism $\widetilde{\eta }: \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ which carries each object $X \in \operatorname{\mathcal{C}}$ to a $\pi $-cocartesian morphism $(X,0) \rightarrow (L(X),1)$ in $\operatorname{\mathcal{E}}$. Composing with the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, we obtain a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor. $\square$
Proposition 6.2.2.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion map. Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a functor of $\infty $-categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \iota \circ L$ be a natural transformation. The following conditions are equivalent:
The natural transformation $\eta $ is the unit of an adjunction: that is, it exhibits $L$ as a left adjoint to the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$.
The natural transformation $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor: that is, for every object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _{X}: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$.
For every object $X \in \operatorname{\mathcal{C}}$, the morphism $L( \eta _ X ): L(X) \rightarrow L(L(X))$ is an isomorphism in $\operatorname{\mathcal{C}}'$. Moreover, if $X$ belongs to $\operatorname{\mathcal{C}}'$, then $\eta _{X}: X \rightarrow L(X)$ is an isomorphism.
Moreover, if these conditions are satisfied, then any natural transformation $\epsilon : L \circ \iota \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}'}$ which is compatible with $\eta $ up to homotopy (in the sense of Definition 6.2.1.1) is an isomorphism in the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}', \operatorname{\mathcal{C}}')$.
Proof. We first show that $(1)$ implies $(2)$. Let $X$ be an object of $\operatorname{\mathcal{C}}$, so that $\eta $ determines a morphism $\eta _{X}: X \rightarrow L(X)$. For every object $Y \in \operatorname{\mathcal{C}}'$, Proposition 6.2.1.17 guarantees that composition with the homotopy class $[\eta _ X]$ induces an isomorphism
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( L(X), Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( L(X), Y) \xrightarrow { \circ [ \eta _ X ] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Y) \]
in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. It follows that $\eta _{X}$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. Allowing $X$ to vary, we conclude that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor.
We now show that $(2)$ implies $(3)$. Assume that, for every object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _ X: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. Note that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{\eta _ X} \ar [d]^{\eta _ X} & L(X) \ar [d]^{ \eta _{L(X) }} \\ L(X) \ar [r]^-{ L( \eta _ X ) } & L(L(X)) } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, obtained by applying the natural transformation $\eta $ to the morphism $\eta _{X}: X \rightarrow L(X)$. For each object $Y \in \operatorname{\mathcal{C}}$, we obtain a commutative diagram of sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Y) & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( L(X), Y) \ar [l]_{ \circ [\eta _ X] } \\ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(L(X), Y) \ar [u]_{ \circ [ \eta _ X ] } & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( L(L(X)), Y). \ar [u]_{ \circ [ \eta _{L(X)} ] } \ar [l]_{ \circ [ L(\eta _ X) ] } } \]
If $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$, then the vertical maps and the upper horizontal map in this diagram are bijective. It follows that the lower horizontal map is bijective as well. Allowing $Y$ to vary, we deduce that the homotopy class $[ L(\eta _ X) ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}'$, so that $L( \eta _ X )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}'$. In the special case where $X$ belongs to $\operatorname{\mathcal{C}}'$, Example 6.2.2.9 guarantees that $\eta _ X$ is already an isomorphism before applying the functor $L$.
We now show that $(3)$ implies $(1)$. Note that $\eta $ determines natural transformations
\[ \eta ': L \rightarrow L \circ \iota \circ L \quad \quad (X \in \operatorname{\mathcal{C}}) \mapsto ( L(\eta _ X) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(L(X), L(L(X)) ) ) \]
\[ \eta '': \iota \rightarrow \iota \circ L \circ \iota \quad \quad (Y \in \operatorname{\mathcal{C}}') \mapsto (\eta _ Y \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, L(Y) ) ). \]
If condition $(3)$ is satisfied, then Theorem 4.4.4.4 guarantees that $\eta '$ and $\eta ''$ are isomorphisms in the $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}' )$ and $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{C}})$, respectively. Invoking the criterion of Proposition 6.1.4.6, we conclude that $\eta $ is the unit of an adjunction. $\square$
Proof of Proposition 6.2.2.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. It follows from Proposition 6.2.2.23 that the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint if and only if there exists a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor. By virtue of Lemma 6.2.2.22, this is equivalent to the requirement that $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$. The analogous characterization of coreflective subcategories follows by a similar argument. $\square$
Example 6.2.2.24. Combining Example 6.2.2.11 with Proposition 6.2.2.18, we see that the inclusion functor $\operatorname{\mathcal{S}}\hookrightarrow \operatorname{\mathcal{QC}}$ admits both a right adjoint (given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$) and a left adjoint (given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ex}^{\infty }(\operatorname{\mathcal{C}})$).
Corollary 6.2.2.25. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
The functor $G$ is fully faithful and the essential image of $G$ is a reflective subcategory of $\operatorname{\mathcal{C}}$.
The functor $G$ is fully faithful and admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.
There exist a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural isomorphism $\epsilon : F \circ G \xrightarrow {\sim } \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is the counit of an adjunction between $F$ and $G$.
The functor $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ for which the composition $(F \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories.
Proof. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the essential image of $G$. If $G$ is fully faithful, then it induces an equivalence $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}'$ (Corollary 4.6.2.23). The equivalence $(1) \Leftrightarrow (2)$ follows by applying Proposition 6.2.2.18 to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$, and the implication $(2) \Rightarrow (3)$ follows by applying Proposition 6.2.2.23 to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. To show that $(3) \Rightarrow (2)$, we observe that if a natural isomorphism $\epsilon : F \circ G \xrightarrow {\sim } \operatorname{id}_{\operatorname{\mathcal{D}}}$ is the counit of an adjunction, then $G$ restricts to an equivalence of $\operatorname{\mathcal{D}}$ with a full subcategory of $\operatorname{\mathcal{C}}$ (Proposition 6.2.1.13), and is therefore fully faithful. The equivalence $(3) \Leftrightarrow (4)$ is a special case of Proposition 6.1.4.7. $\square$
Remark 6.2.2.26. In the situation of Corollary 6.2.2.25, suppose that $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is the unit of an adjunction between $F$ and $G$. Then an object $C \in \operatorname{\mathcal{C}}$ belongs to the essential image of $G$ if and only if the unit map $\eta _{C}: C \rightarrow (G \circ F)(C)$ is an isomorphism. The “if” direction is obvious. To prove the converse, we may assume without loss of generality that $C = G(D)$, for some object $D \in \operatorname{\mathcal{D}}$. In this case, the morphism $\eta _{C} = \eta _{G(D)}$ fits into a commutative diagram where $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is compatible with $\epsilon $ up to homotopy. Since $\epsilon $ is an isomorphism, it follows that $\eta _{C} = \eta _{G(D)}$ is also an isomorphism.
\[ \xymatrix@C =50pt@R=50pt{ & (G \circ F \circ G)(D) \ar [dr]^{ G( \epsilon _{D} ) } & \\ G(D) \ar [ur]^{ \eta _{G(D)} } \ar [rr]^{\operatorname{id}_{G(D)}} & & G(D), } \]
Corollary 6.2.2.27. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is an equivalence if and only if it satisfies the following pair of conditions:
The functor $F$ is conservative. That is, a morphism $u$ of $\operatorname{\mathcal{C}}$ is an isomorphism if and only if $F(u)$ is an isomorphism in $\operatorname{\mathcal{D}}$.
The functor $F$ admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.
Proof. Suppose that conditions $(1)$ and $(2)$ are satisfied; we will show that $F$ is an equivalence of $\infty $-categories (the converse is immediate from the definitions). Combining assumption $(2)$ with Corollary 6.2.2.25, we can choose a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural isomorphism $\epsilon : F \circ G \xrightarrow {\sim } \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is the counit of an adjunction between $F$ and $G$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation which is compatible up to homotopy with $\epsilon $, in the sense of Definition 6.2.1.1. For each object $C \in \operatorname{\mathcal{C}}$, the morphism $\eta _{C}$ fits into a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ & (F \circ G \circ F)(C) \ar [dr]^{ \epsilon _{F(C)} } & \\ F(C) \ar [ur]^{ F( \eta _ C ) } \ar [rr]^{\operatorname{id}_{F(C)}} & & F(C) } \]
in the $\infty $-category $\operatorname{\mathcal{D}}$, where $\epsilon _{ F(C) }$ and $\operatorname{id}_{ F(C) }$ are isomorphisms. It follows that $F( \eta _ C )$ is also an isomorphism in $\operatorname{\mathcal{D}}$. Applying assumption $(1)$, we deduce that $\eta _{C}$ is an isomorphism in $\operatorname{\mathcal{C}}$. Allowing the object $C$ to vary (and invoking the criterion of Theorem 4.4.4.4), we deduce that $\eta $ is also a natural isomorphism, so that $F$ and $G$ are homotopy inverse to one another. $\square$
Corollary 6.2.2.28. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $L$ be a functor from $\operatorname{\mathcal{C}}$ to itself, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ be a natural transformation. The following conditions are equivalent:
For every object $X \in \operatorname{\mathcal{C}}$, the morphisms $L(\eta _ X): L(X) \rightarrow L(L(X))$ and $\eta _{L(X)}: L(X) \rightarrow L(L(X))$ are isomorphisms.
There exists a full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ for which $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor, in the sense of Definition 6.2.2.19.
Proof. The implication $(2) \Rightarrow (1)$ follows from Proposition 6.2.2.23. Conversely, suppose that condition $(1)$ is satisfied, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects of the form $L(X)$ for $X \in \operatorname{\mathcal{C}}$. Assumption $(1)$ guarantees that $\eta _{Y}$ is an isomorphism for each $Y \in \operatorname{\mathcal{C}}'$, so that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor by virtue of Proposition 6.2.2.23. $\square$
Exercise 6.2.2.29. Suppose that the conditions of Corollary 6.2.2.28 are satisfied and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory of $\operatorname{\mathcal{C}}$. Show that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor if and only if the following conditions are satisfied:
For each object $X \in \operatorname{\mathcal{C}}$, the object $L(X)$ is contained in $\operatorname{\mathcal{C}}'$.
For each object $Y \in \operatorname{\mathcal{C}}'$, there exists an isomorphism $Y \rightarrow L(X)$ for some object $X \in \operatorname{\mathcal{C}}$.
If the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is replete (Example 4.4.1.12), then it is uniquely determined by these conditions.