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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.

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 $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ if $Y$ belongs to $\operatorname{\mathcal{C}}'$ and, for every object $Z \in \operatorname{\mathcal{C}}'$, the precomposition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [u] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We say that $u$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-colocalization of $Y$ if $X$ belongs to $\operatorname{\mathcal{C}}'$ and, for every object $W \in \operatorname{\mathcal{C}}'$, the postcomposition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \xrightarrow { [u] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

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 if 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 6.2.2.2. 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 6.2.2.3. 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 $X$. Stated more informally: a $\operatorname{\mathcal{C}}'$-reflection of $X$, if it exists, is unique up to isomorphism.

Example 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 $X$ 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.

Our first goal is to prove the following:

Proposition 6.2.2.5. 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 6.2.2.5 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 6.2.2.6. 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 6.2.2.1. 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 6.2.2.7. In the situation of Definition 6.2.2.6, 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 $\operatorname{\mathcal{C}}'$.

Lemma 6.2.2.8. 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 $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ 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 5.1.2.4, the morphism $\widetilde{u}$ is $\pi $-cocartesian if and only if $u$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-localization 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 5.2.2.4, 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 6.2.2.9. 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:

$(1)$

The natural transformation $\eta $ is the unit of an adjunction: that is, it exhibits $L$ as a left adjoint to the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$.

$(2)$

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 $X$.

$(3)$

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 6.2.2.4 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 6.2.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. It follows from Proposition 6.2.2.9 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 6.2.2.8, 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. $\square$

Corollary 6.2.2.10. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. The following conditions are equivalent:

$(1)$

The functor $G$ is fully faithful and the essential image of $G$ is a reflective subcategory of $\operatorname{\mathcal{C}}$.

$(2)$

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

$(3)$

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 $G$.

$(4)$

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 4.6.2.19). The equivalence $(1) \Leftrightarrow (2)$ follows by applying Proposition 6.2.2.5 to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$, and the implication $(2) \Rightarrow (3)$ follows by applying Proposition 6.2.2.9 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. $\square$

Corollary 6.2.2.11. 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:

$(1)$

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.

$(2)$

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 6.2.2.6.

Proof. The implication $(2) \Rightarrow (1)$ follows from Proposition 6.2.2.9. 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 6.2.2.9. $\square$

Exercise 6.2.2.12. In the situation of Corollary 6.2.2.11, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory of $\operatorname{\mathcal{C}}$. Show that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-localization 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 4.4.1.11), then it is uniquely determined by these conditions.