# Kerodon

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### 6.3.3 Reflective Localizations

It will often be convenient to work with the following variant of Definition 6.3.1.9.

Definition 6.3.3.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. We will say that $F$ is a localization functor if there exists a collection of morphisms $W$ in $\operatorname{\mathcal{C}}$ such that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

In the situation of Definition 6.3.3.1, there is always a canonical choice for the collection of morphisms $W$:

Proposition 6.3.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $W$ be the collection of all morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $F$ is a localization functor if and only if it exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

Proof. Assume that $F$ is a localization functor; we will show that it exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (the reverse implication follows immediately from the definitions). Let $\operatorname{\mathcal{E}}$ be an $\infty$-category; we wish to show that composition with $F$ induces an equivalence of $\infty$-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$. Choose a collection of morphisms $W'$ of $\operatorname{\mathcal{C}}$ such that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W'$. Note that $W'$ is contained in $W$, so that we can regard $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}})$ as a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}[W'^{-1}], \operatorname{\mathcal{D}})$. It will therefore suffice to show that the composite functor

$\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \hookrightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W'^{-1}], \operatorname{\mathcal{E}})$

is an equivalence of $\infty$-categories, which follows from our assumption on the functor $F$. $\square$

We now use the ideas of §6.2.2 to describe a large class of localization functors.

Definition 6.3.3.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. We say that $F$ is a reflective localization functor if it admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which is fully faithful.

Remark 6.3.3.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Using Corollary 6.2.2.17 to the right adjoint of $F$, we see that the following conditions are equivalent:

• The functor $F$ is a reflective localization: that is, it admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

• There exists 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 $G$.

• The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty$-categories.

Moreover, if these conditions are satisfied, then the essential image of $G$ is a reflective subcategory of $\operatorname{\mathcal{C}}$.

Remark 6.3.3.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a reflective localization of $\infty$-categories. Then there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which is the unit of an adjunction between $F$ and $G$. The natural transformation $\eta$ is generally not an isomorphism (unless $F$ is an equivalence of $\infty$-categories). However, our assumption that $F$ is a reflective localization guarantees that there exists a natural isomorphism $\epsilon : F \circ G \xrightarrow {\sim } \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is compatible with $\eta$ up to homotopy. In particular, for every object $X \in \operatorname{\mathcal{C}}$, the composition

$F(X) \xrightarrow { F( \eta _ X ) } (F \circ G \circ F)(X) \xrightarrow { \epsilon _{F(X)} } F(X)$

is homotopic to the identity $\operatorname{id}_{ F(X) }$, which guarantees that $F( \eta _ X )$ is an isomorphism in $\operatorname{\mathcal{D}}$.

Proposition 6.3.3.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

The functor $F$ is a reflective localization (in the sense of Definition 6.3.3.3): that is, it admits a fully faithful right adjoint.

$(2)$

The functor $F$ is a localization functor (in the sense of Definition 6.3.3.1) which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

Proof. We first show that $(2)$ implies $(1)$. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a right adjoint to $F$ and let $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be the counit of an adjunction. We wish to show that, if $F$ is a localization functor, then $\epsilon$ is an isomorphism (Remark 6.3.3.4). By virtue of Proposition 6.1.4.7 (applied to the opposite of the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$), it will suffice to show that for any $\infty$-category $\operatorname{\mathcal{E}}$, precomposition with the isomorphism class $[F] \in \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )$ induces a monomorphism

$\pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \xrightarrow {\circ [F]} \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ),$

which follows immediately from our assumption on $F$.

We now prove the converse. Assume that $F$ is a reflective localization functor and let $W$ be the collection of all morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$; we will show that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Fix an $\infty$-category $\operatorname{\mathcal{E}}$, so that precomposition with $F$ induces a function

$\varphi : \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ).$

Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be as above, so that precomposition with $G$ induces a function

$\psi : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ) \subseteq \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ).$

We will show that $\psi$ is inverse to $\varphi$. By assumption, the natural transformation $\epsilon$ is an isomorphism. It follows that, for every functor $E: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, $\epsilon$ induces an isomorphism $E \circ F \circ G \xrightarrow {\sim } E$. Passing to isomorphism classes, we obtain an equality $[E] = [E \circ F \circ G] = \psi ( [ E \circ F ] ) = (\psi \circ \varphi )([E])$. Allowing $E$ to vary, we conclude that $\psi \circ \varphi$ is the identity on $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } )$.

We now complete the proof by showing that $\varphi \circ \psi$ is the identity on $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } )$. Let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories which carries each morphism of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$; we wish to show that $H$ is isomorphic to $H \circ G \circ F$. Choose a unit map $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which is compatible with $\epsilon$ up to homotopy. For every object $C \in \operatorname{\mathcal{C}}$, Remark 6.3.3.5 guarantees that the morphism $\eta _{C}: C \rightarrow (G \circ F)(C)$ belongs to $W$, so that $H(\eta _{C} )$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{E}}$. Allowing the object $C$ to vary (and invoking the criterion of Theorem 4.4.4.4), we conclude that $\eta$ induces an isomorphism from $H$ to $H \circ G \circ F$ in the functor $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. $\square$

Example 6.3.3.7. 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 (Definition 6.2.2.12). Then $L$ is a reflective localization functor (since it is left adjoint to the inclusion map $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$; see Proposition 6.2.2.15). In particular, $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to the collection of morphisms $w$ such that $L(w)$ is an isomorphism.

Remark 6.3.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Up to equivalence, every reflective localization functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ can be obtained from the construction of Example 6.3.3.7. More precisely, if $G$ is a fully faithful right adjoint to $F$, then it induces an equivalence of $\operatorname{\mathcal{D}}$ with a reflective subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ (carrying $F$ to the $\operatorname{\mathcal{C}}'$-reflection functor $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$); see Corollary 6.2.2.17.

Remark 6.3.3.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, where $F$ is a reflective localization functor. Then $G$ is a reflective localization functor if and only if $(G \circ F)$ is a reflective localization functor. In particular, the collection of reflective localization functors is closed under composition. See Remarks 6.2.1.8 and 4.6.2.5.

Warning 6.3.3.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories. If $F$ and $G$ are localization functors, the composition $G \circ F$ need not be a localization functor. For example, let $\operatorname{\mathcal{C}}$ be the $1$-dimensional simplicial set corresponding to the directed graph depicted in the diagram

$\bullet \xrightarrow {e} \bullet \xleftarrow {w} \bullet \xrightarrow {e'} \bullet .$

There is a functor $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^2$ which carries $e$ and $e'$ to the edges $0 \rightarrow 1$ and $1 \rightarrow 2$, respectively. It is not difficult to show that $F$ exhibits $\Delta ^2$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $w$. However, the localization of $\Delta ^2$ with respect to its “long edge” $0 \rightarrow 2$ cannot be realized directly as a localization of $\operatorname{\mathcal{C}}$.

Variant 6.3.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. We say that $F$ is a coreflective localization functor if it admits a fully faithful left adjoint $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Equivalently, $F$ is a coreflective localization functor if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is a reflective localization functor (see Remark 6.2.1.7). It follows from Proposition 6.3.3.6 that every coreflective localization functor is a localization functor (in the sense of Definition 6.3.3.1).

Warning 6.3.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a localization functor of $\infty$-categories which is both reflective and coreflective. Then $F$ admits both a left adjoint $F^{L}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a right adjoint $F^{R}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, which are automatically fully faithful (Proposition 6.3.3.6). Let $\operatorname{\mathcal{C}}^{L} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the essential image of the functor $F^{L}$, and define $\operatorname{\mathcal{C}}^{R} \subseteq \operatorname{\mathcal{C}}$ similarly. When viewed as abstract $\infty$-categories, $\operatorname{\mathcal{C}}^{L}$ and $\operatorname{\mathcal{C}}^{R}$ are equivalent (since they are both equivalent to the $\infty$-category $\operatorname{\mathcal{D}}$). Beware that they generally do not coincide as subcategories of $\operatorname{\mathcal{C}}$. See Warning 9.1.1.23.