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6.3.3 Reflective Localizations

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. In §6.3.2, we proved that there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as the localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Proposition 6.3.2.1). The construction of §6.3.2 was fairly inexplicit, and gave little information about the structure of the localization $\operatorname{\mathcal{C}}[W^{-1}]$ other than its universal property. In this section, we study a special class of localizations which can be described more concretely, by identifying them with reflective (or coreflective) subcategories of $\operatorname{\mathcal{C}}$.

Definition 6.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that an object $Z \in \operatorname{\mathcal{C}}$ is $W$-local if, for every morphism $w: X \rightarrow Y$ belonging to $W$, precomposition with the homotopy class $[w]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We say that $Z$ is $W$-colocal if, for every morphism $w: Y \rightarrow X$ belonging to $W$, postcomposition with the homotopy class $[w]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Z, Y) \xrightarrow { [w] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z,X)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Definition 6.3.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $W$ is localizing if the following conditions are satisfied:

$(1)$

Every isomorphism of $\operatorname{\mathcal{C}}$ is contained in $W$.

$(2)$

The collection of morphisms $W$ satisfies the two-out-of-three property. That is, for every $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]

of $\operatorname{\mathcal{C}}$, if any two of the morphisms $u$, $v$, and $w$ belong to $W$, then so does the third.

$(3)$

For every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $w: Y \rightarrow Z$ which belongs to $W$, where the object $Z$ is $W$-local.

We say that $W$ is colocalizing if it satisfies conditions $(1)$ and $(2)$ together with the following dual version of $(3)$:

$(3')$

For every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $w: X \rightarrow Y$ which belongs to $W$, where $X$ is $W$-colocal.

Remark 6.3.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, which we also view as a collection of morphisms in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then an object $Z \in \operatorname{\mathcal{C}}$ is $W$-local (in the sense of Definition 6.3.3.1) if and only if it is $W$-colocal when viewed as an object of $\operatorname{\mathcal{C}}^{\operatorname{op}}$. The collection of morphisms $W$ is localizing (in the sense of Definition 6.3.3.2) if and only if it is colocalizing when viewed as a collection of morphisms of $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Remark 6.3.3.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $w: X \rightarrow Y$ be a morphism which belongs to $W$. Then, for every $W$-local object $Z$ of $\operatorname{\mathcal{C}}$, precomposition with the homotopy class $[w]$ induces a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y, Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z)$. In particular, if the objects $X$ and $Y$ are $W$-local, then $w$ is an isomorphism.

Proposition 6.3.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is localizing, and let $\operatorname{\mathcal{C}}'$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. Then:

$(1)$

The full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective (Definition 6.2.2.1).

$(2)$

Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a left adjoint to the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$. Then $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, in the sense of Definition 6.3.1.9.

$(3)$

A morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ belongs to $W$ if and only if $L(f)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}'$.

Proof. Let $X$ be an object of $\operatorname{\mathcal{C}}$. Our assumption that $W$ is localizing guarantees that there exists a morphism $w: X \rightarrow Y$ which belongs to $W$, where $Y$ is $W$-local. Note that, if $Z$ is any $W$-local object of $\operatorname{\mathcal{C}}$, then composition with the homotopy class $[w]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. It follows that $w$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, in the sense of Definition 6.2.2.1. This proves $(1)$.

It follows from $(1)$ that the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint (Proposition 6.2.2.7). Choose a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \iota \circ L$ which is the unit of an adjunction between $L$ and $\iota $. Then, 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 $X$ (Proposition 6.2.2.11). Since $W$ is localizing, we can also choose a morphism $w: X \rightarrow Y$ of $W$, where $Y$ belongs to $\operatorname{\mathcal{C}}'$. Arguing as above, we see that $w$ also exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. It follows from Remark 6.2.2.3 that we can realize $\eta _{X}$ as a composition of $w$ with an isomorphism $Y \rightarrow L(X)$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $W$ contains all isomorphisms and is closed under composition, it follows that $\eta _{X}$ also belongs to $W$.

We now prove $(3)$. Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We then have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{\eta _ X} & Y \ar [d]^{\eta _ Y} \\ L(X) \ar [r]^-{ L(f) } & L(Y), } \]

where $\eta _ X$ and $\eta _ Y$ belong to $W$. Since $W$ satisfies the two-out-of-three property, it follows that $f$ belongs to $W$ if and only if $L(f)$ belongs to $W$. Since $L(X)$ and $L(Y)$ are $W$-local objects of $\operatorname{\mathcal{C}}$, this is equivalent to the requirement that $L(f)$ is an isomorphism (Remark 6.3.3.4).

We now prove $(2)$ using the criterion of Proposition 6.3.1.13. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those functors which carry each morphism of $W$ to an isomorphism of $\operatorname{\mathcal{E}}$ (Notation 6.3.1.1). It follows from $(2)$ that precomposition with the functor $L$ induces a map

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

We wish to show that $\theta $ is bijective. To prove injectivity, we observe that the construction $[F] \mapsto [F|_{\operatorname{\mathcal{C}}} ]$ determines a left inverse to $\theta $ (any functor $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{E}}$ is isomorphic to the restriction $(F' \circ L)|_{\operatorname{\mathcal{C}}'}$ via the natural transformation $\eta $). To prove surjectivity, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be any functor. Then $\eta $ induces a natural transformation $\eta ': F \rightarrow F|_{\operatorname{\mathcal{C}}'} \circ L$, which carries each object $X \in \operatorname{\mathcal{C}}$ to the morphism $F(\eta _ X): F(X) \rightarrow (F \circ L)(X)$. If $F$ carries each morphism of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$, then $\eta '$ is a natural isomorphism (Theorem 4.4.4.4). In particular, $F$ is isomorphic to $F|_{\operatorname{\mathcal{C}}'} \circ L$ so that the isomorphism class $[F]$ belongs to the image of $\theta $. $\square$

Notation 6.3.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a localizing collection of morphisms of $\operatorname{\mathcal{C}}$. We will often write $\operatorname{\mathcal{C}}[W^{-1}]$ for the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. By virtue of Proposition 6.3.3.5, this is consistent with Remark 6.3.2.2: that is, we can regard $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. This convention is very convenient, since the full subcategory of $W$-local objects is uniquely determined by $\operatorname{\mathcal{C}}$ and $W$. However, it has the potential to create confusion in some situations: see Warning 6.3.3.8 below.

Proposition 6.3.3.5 has a counterpart for colocalizing collections of morphisms:

Variant 6.3.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is colocalizing, and let $\operatorname{\mathcal{C}}'$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-colocal objects. Then:

$(1)$

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

$(2)$

Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a right adjoint to the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$. Then $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

$(3)$

A morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ belongs to $W$ if and only if $L(f)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}'$.

Warning 6.3.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is both localizing and colocalizing. In this case, Proposition 6.3.3.5 and Variant 6.3.3.7 provide two different concrete realizations of the localization $\operatorname{\mathcal{C}}[W^{-1}]$, given by the full subcategories $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}\supseteq \operatorname{\mathcal{C}}''$ spanned by the $W$-local and $W$-colocal objects of $\operatorname{\mathcal{C}}$, respectively. Note that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ are necessarily equivalent as abstract $\infty $-categories. More precisely, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ is a functor which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as the localization of $\operatorname{\mathcal{C}}$ with respect to $W$, then the restrictions

\[ \operatorname{\mathcal{C}}' \xrightarrow { F|_{\operatorname{\mathcal{C}}'} } \operatorname{\mathcal{C}}[W^{-1}] \xleftarrow { F|_{\operatorname{\mathcal{C}}''} } \operatorname{\mathcal{C}}'' \]

are equivalences of $\infty $-categories. Beware that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ usually do not coincide when regarded as subcategories of $\operatorname{\mathcal{C}}$.

Proposition 6.3.3.5 admits a converse:

Proposition 6.3.3.9. 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 left adjoint to the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$. Let $W$ be the collection of all morphisms $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ for which $L(f)$ is an isomorphism in $\operatorname{\mathcal{C}}'$. Then:

$(1)$

The collection $W$ is localizing (Definition 6.3.3.2).

$(2)$

Every object of $\operatorname{\mathcal{C}}'$ is $W$-local (Definition 6.3.3.1).

$(3)$

If $\operatorname{\mathcal{C}}'$ is replete, then every $W$-local object of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}'$.

Proof. We first prove $(2)$. Let $Z$ be an object of $\operatorname{\mathcal{C}}'$ and let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which belongs to $W$; we wish to show that precomposition with the homotopy class $[w]$ induces an isomorphism

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Using Proposition 6.2.1.17, we can identify $\theta $ with the map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( L(Y), Z) \xrightarrow { \circ [L(w)] } \operatorname{Hom}{\operatorname{\mathcal{C}}'}( L(X), Z), \]

which is invertible by virtue of our assumption that $L(w)$ is an isomorphism of $\operatorname{\mathcal{C}}'$.

We now prove $(1)$. It follows immediately from the definitions that $W$ contains all isomorphisms of $\operatorname{\mathcal{C}}$ and satisfies the two-out-of-three property. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \iota \circ L$ be the unit of an adjunction. Then $\eta $ carries each object $X \in \operatorname{\mathcal{C}}$ to a morphism $\eta _ X: X \rightarrow L(X)$, where $L(X)$ belongs to $\operatorname{\mathcal{C}}'$ and is therefore $W$-local (by virtue of $(2)$). Moreover, $L(\eta _ X)$ is an isomorphism in $\operatorname{\mathcal{C}}'$ (Proposition 6.2.2.11), so $\eta _ X$ belongs to $W$.

We now prove $(3)$. Suppose that $X$ is a $W$-local object of $\operatorname{\mathcal{C}}$. Then $\eta _ X: X \rightarrow L(X)$ is a morphism between $W$-local objects of $\operatorname{\mathcal{C}}$. Since $\eta _ X$ belongs to $W$, it follows that $\eta _ X$ is an isomorphism (Remark 6.3.3.4). If the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is replete, we conclude that $X$ belongs to $\operatorname{\mathcal{C}}'$. $\square$

Corollary 6.3.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then there is a canonical bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Localizing collections of morphisms of $\operatorname{\mathcal{C}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Reflective replete subcategories of $\operatorname{\mathcal{C}}$} \} , } \]

which carries a localizing collection of morphisms $W$ to the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ spanned by the $W$-local objects.

Proof. Combine Proposition 6.3.3.5 with Proposition 6.3.3.9. $\square$

Definition 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 reflective localization if it exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, where $W$ is a localizing collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $F$ is a coreflective localization if it exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, where $W$ is a colocalizing collection of morphisms of $\operatorname{\mathcal{C}}$.

Remark 6.3.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a reflective localization functor. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to some localizing collection of morphisms $W$. The collection $W$ is then uniquely determined: it is the collection of all morphisms $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ for which $F(u)$ is an isomorphism of $\operatorname{\mathcal{D}}$. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}[W^{-1}]$ is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects and that $F$ is a left adjoint to the inclusion functor $\operatorname{\mathcal{C}}[W^{-1}] \hookrightarrow \operatorname{\mathcal{C}}$, in which case it follows from Proposition 6.3.3.5.

Reflective localization functors admit many characterizations:

Proposition 6.3.3.13. 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.

$(2)$

The functor $F$ admits a right adjoint and exhibits $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $W$ of $\operatorname{\mathcal{C}}$.

$(3)$

The functor $F$ admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

$(4)$

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 exhibits $G$ as a right adjoint of $F$.

$(5)$

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

Proof. Note that any of these conditions guarantee that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. The equivalence $(1) \Leftrightarrow (3) \Leftrightarrow (4) \Leftrightarrow (5)$ follow by applying Corollary 6.2.2.13 to the functor $G$, and the implication $(1) \Rightarrow (2)$ is immediate. We will complete the proof by showing that $(2)$ implies $(4)$. Assume that $F$ exhibits $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to a collection of morphisms $W$, and let $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be the counit of an adjunction. We wish to show that $\epsilon $ is an isomorphism. 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 $F$. $\square$