nLab reflective localization

Idea

A localization of a category/of an (∞,1)-category is called reflective if its localization functor has a fully faithful right adjoint, hence if it is the reflector of a reflective subcategory/reflective sub-(∞,1)-category-inclusion.

In fact every reflective subcategory inclusion exhibits a reflective localization (Prop. below).

For reflective localizations the localized category has a particularly useful description (Prop. below): It is equivalent to the full subcategory of local objects (Def. below).

Therefore, sometimes reflective localizations at a class $S$ of morphisms are understood as the default concept of localization, in fact often reflection onto the full subcategory of $S$-local objects (Def. below) is understood by default. Notably left Bousfield localizations are presentations of reflective localizations of (∞,1)-categories in this sense.

These reflections onto $S$-local objects satisfy the universal property of an $S$-localization (only) for all left adjoint functors that invert the class $S$ (Prop. below).

Definition

Reflective localization

Definition

(category with weak equivalences)

A category with weak equivalences is

1. a category $\mathcal{C}$,

2. a subcategory $W \subset \mathcal{C}$

such that the morphisms in $W$

1. include all the isomorphisms of $\mathcal{C}$,

2. satisfy two-out-of-three:

   If for $g$, $f$ any two composable morphisms in $\mathcal{C}$, two out of the set $\{g,\, f,\, g \circ f \}$ are in $W$, then so is the third.

   $$\array{ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ && \underset{ g \circ f }{\longrightarrow} }$$

Definition

(localization of a category)

Let $W \subset \mathcal{C}$ be a category with weak equivalences (Def. ). Then the localization of $\mathcal{C}$ at $W$ is, if it exists

1. a category $\mathcal{C}[W^{-1}]$

2. a functor $\gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]$

such that

1. $\gamma$ sends all morphisms in $W \subset \mathcal{C}$ to isomorphisms,

2. $\gamma$ is universal with this property: If $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ is any functor with this property, then it factors through $\gamma$, up to natural isomorphism:

   $$F \;\simeq\; D F \circ \gamma \phantom{AAAAAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D F}} \\ && \mathcal{C}[W^{-1}] }$$

   and any two such factorizations $D F$ and $D^' F$ are related by a unique natural isomorphism $\kappa$ compatible with $\rho$ and $\rho^'$:

(1)$$\array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D F}} && \searrow^{\mathrlap{id}} \\ && \mathcal{C}[W^{-1}] && {}_{\simeq}\seArrow^{\kappa} && \mathcal{D} \\ && & {}_{\mathllap{id}}\searrow && \nearrow_{\mathrlap{D^' F}} \\ && && \mathcal{C}[W^{-1}] } \phantom{AAAA} = \phantom{AAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho^'}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D^' F}} \\ && \mathcal{C}[W^{-1}] }$$

Such a localization is called a reflective localization if the localization functor has a fully faithful right adjoint, exhibiting it as the reflection functor of a reflective subcategory-inclusion

$$\mathcal{C}[W^{-1}] \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{ \phantom{AA} \gamma \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C}$$

Reflection onto local objects

It turns out (Prop. ) below, that reflective localizations at a collection $S$ of morphisms are, when they exist, reflections onto the full subcategory of $S$-local objects (Def. below). Often this reflection of $S$-local objects is what one is more interested in than the universal property of the $S$-localization according to (Def. ). This reflection onto local objects (Def. below) is what is often meant by default with “localization” (for instance in Bousfield localization).

Definition

(local object)

Let $\mathcal{C}$ be a category and let $S \subset Mor_{\mathcal{C}}$ be a set of morphisms. Then an object $X \in \mathcal{C}$ is called an $S$-local object if for all $A \overset{s}{\to} B \; \in S$ the hom-functor from $s$ into $X$ yields a bijection

$$Hom_{\mathcal{C}}(s,X) \;\colon\; Hom_{\mathcal{C}}(B,X) \overset{ \phantom{AA} \simeq \phantom{AA} }{\longrightarrow} Hom_{\mathcal{C}}(A,X) \,,$$

hence if every morphism $A \overset{f}{\longrightarrow} X$ extends uniquely along $s$ to $B$:

$$\array{ A &\overset{\phantom{A}f\phantom{A}}{\longrightarrow}& X \\ {}^{\mathllap{s}}\big\downarrow & \nearrow_{\mathrlap{ \exists! }} \\ B }$$

We write

(2)$$\mathcal{C}_S \overset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow} \mathcal{C}$$

for the full subcategory of $S$-local objects.

Definition

(reflection onto full subcategory of local objects)

Let $\mathcal{C}$ be a category and set $S \subset Mor_{\mathcal{C}}$ be a sub-class of its morphisms. Then the reflection onto local $S$-objects (often called “localization at the collection $S$”) is, if it exists, a left adjoint $L$ to the full subcategory-inclusion of the $S$-local objects (2):

$$\mathcal{C}_S \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{C} \,.$$

Properties

Proposition

(reflective subcategories are localizations)

Every reflective subcategory-inclusion

$$\mathcal{C}_{L} \underoverset {\underset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}} {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C}$$

is the reflective localization at the class $W \coloneqq L^{-1}(Isos)$ of morphisms that are sent to isomorphisms by the reflector $L$.

Proof

Let $F \;\colon\; \mathcal{C} \to \mathcal{D}$ be a functor which inverts morphisms that are inverted by $L$.

First we need to show that it factors through $L$, up to natural isomorphism. But consider the following whiskering $F(\eta)$ of the adjunction unit $\eta$ with $F$:

$$\array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{L}}\searrow &\Downarrow& \nearrow_{\mathrlap{D F}} \\ && \mathcal{C}_L } \phantom{AA} \coloneqq \phantom{AA} \array{ \mathcal{C} && \overset{id}{\longrightarrow} && \mathcal{C} & \overset{F}{\longrightarrow}& \mathcal{D} \\ & {}_{\mathllap{L}}\searrow &\Downarrow^{\eta}& \nearrow_{\mathrlap{\iota}} \\ && \mathcal{C}_L }$$

By idempotency, the components of the adjunction unit $\eta$ are inverted by $L$, and hence by assumption they are also inverted by $F$, so that on the right the natural transformation $F(\eta)$ is indeed a natural isomorphism.

It remains to show that this factorization is unique up to unique natural isomorphism. So consider any other factorization $D^' F$ via a natural isomorphism $\rho$. Pasting this now with the adjunction counit

$$\array{ && \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}^{\mathllap{\iota}}\nearrow & {}^{\epsilon}\Downarrow & {}_{\mathllap{L}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}} \\ \mathcal{C}_L && \underset{ id }{\longrightarrow} && \mathcal{C}_L }$$

exhibits a natural isomorphism $\epsilon \cdot \rho$ between $D F \simeq D^' F$. Moreover, this is compatible with $F(\eta)$ according to (1), due to the triangle identity:

$$\array{ \mathcal{C} && \overset{id}{\longrightarrow} && \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{id}}\searrow & {}^{\mathllap{\eta}}\Downarrow & {}^{\mathllap{\iota}}\nearrow & {}^{\epsilon}\Downarrow & {}_{\mathllap{L}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}} \\ && \mathcal{C}_L && \underset{ id }{\longrightarrow} && \mathcal{C}_L } \phantom{AAAA} = \phantom{AAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & \searrow &\Downarrow^\rho& \swarrow \\ && \mathcal{C}_L }$$

Finally, since $L$ is essentially surjective functor, by idempotency, it is clear that this is the unique such natural isomorphism.

Proposition

(reflective localization reflects onto full subcategory of local objects)

Let $W \subset \mathcal{C}$ be a category with weak equivalences (Def. ). If its reflective localization (Def. ) exists

$$\mathcal{C}[W^{-1}] \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C}$$

then $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is equivalently the inclusion of the full subcategory on the $W$-local objects (Def. ), and hence $L$ is equivalently reflection onto the $W$-local objects, according to Def. .

Proof

We need to show that

1. every $X \in \mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is $W$-local,

2. every $Y \in \mathcal{C}$ is $W$-local precisely if it is isomorphic to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.

The first statement follows directly with the adjunction isomorphism:

$$Hom_{\mathcal{C}}(w, \iota(X)) \simeq Hom_{\mathcal{C}[W^{-1}]}(L(w), X)$$

and the fact that the hom-functor takes isomorphisms to bijections.

For the second statement, consider the case that $Y$ is $W$-local. Observe that then $Y$ is also local with respect to the class

$$W_{sat} \;\coloneqq\; L^{-1}(Isos)$$

of all morphisms that are inverted by $L$ (the “saturated class of morphisms”): For consider the hom-functor $\mathcal{C} \overset{Hom_{\mathcal{C}}(-,Y)}{\longrightarrow} Set^{op}$ to the opposite of the category of sets. But assumption on $Y$ this takes elements in $W$ to isomorphisms. Hence, by the defining universal property of the localization-functor $L$, it factors through $L$, up to natural isomorphism.

Since by idempotency the adjunction unit $\eta_Y$ is in $W_{sat}$, this implies that we have a bijection of the form

$$Hom_{\mathcal{C}}( \eta_Y, Y ) \;\colon\; Hom_{\mathcal{C}}( \iota L(Y), Y ) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}(Y, Y) \,.$$

In particular the identity morphism $id_Y$ has a preimage $\eta_Y^{-1}$ under this function, hence a left inverse to $\eta$:

$$\eta_Y^{-1} \circ \eta_Y \;=\; id_Y \,.$$

But by 2-out-of-3 this implies that $\eta_Y^{-1} \in W_{sat}$. Since the first item above shows that $\iota L(Y)$ is $W_{sat}$-local, this allows to apply this same kind of argument again,

$$Hom_{\mathcal{C}}( \eta^{-1}_Y, \iota L(Y) ) \;\colon\; Hom_{\mathcal{C}}( Y, \iota L(Y) ) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}( \iota L(Y) , \iota L(Y)) \,,$$

to deduce that also $\eta_Y^{-1}$ has a left inverse $(\eta_Y^{-1})^{-1} \circ \eta_Y^{-1}$. But since a left inverse that itself has a left inverse is in fact an inverse morphisms (this Lemma), this means that $\eta^{-1}_Y$ is an inverse morphism to $\eta_Y$, hence that $\eta_Y \;\colon\; Y \to \iota L (Y)$ is an isomorphism and hence that $Y$ is isomorphic to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.

Conversely, if there is an isomorphism from $Y$ to a morphism in the image of $\iota$ hence, by the first item, to a $W$-local object, it follows immediatly that also $Y$ is $W$-local, since the hom-functor takes isomorphisms to bijections and since bijections satisfy 2-out-of-3.

$\,$

• For a left exact reflective localization, the class of morphisms that is inverted forms a left-multiplicative system. For the moment see at geometric embedding for details on this.

Proposition

(reflection onto local objects in localization with respect to left adjoints)

Let $\mathcal{C}$ be a category and let $S \subset Mor_{\mathcal{C}}$ be a class of morphisms in $\mathcal{C}$. Then the reflection onto the $S$-local objects (Def. ) satisfies, if it exists, the universal property of a localization of categories (Def. ) with respect to left adjoint functors inverting $S$.

Proof

Write

$$\mathcal{C}_S \underoverset {\underset{ \phantom{AA}\iota\phantom{AA} }{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{C}$$

for the reflective subcategory-inclusion of the $S$-local objects.

Say that a morphism $f$ in $\mathcal{C}$ is an $S$-local morphism if for every $S$-local object $A \in \mathcal{C}$ the hom-functor from $f$ to $A$ yields a bijection $Hom_{\mathcal{C}}(f,A)$. Notice that, by the Yoneda embedding for $\mathcal{C}_S$, the $S$-local morphisms are precisely the morphisms that are taken to isomorphisms by the reflector $L$.

Now let

$$(F \dashv G) \;\colon\; \mathcal{C} \underoverset {\underset{G}{\longleftarrow}} {\overset{ \phantom{AA} F \phantom{AA} }{\longrightarrow}} {\bot} \mathcal{D}$$

be a pair of adjoint functors, such that the left adjoint $F$ inverts the morphisms in $S$. By the adjunction hom-isomorphism it follows that $G$ takes values in $S$-local objects. This in turn implies, now via the Yoneda embedding for $\mathcal{D}$, that $F$ inverts all $S$-local morphisms, and hence all morphisms that are inverted by $L$.

Thus the essentially unique factorization of $F$ through $L$ now follows by Prop. .

Related concepts

• reflective subcategory, coreflective subcategory

• essential localization

  quintessential localization

  bireflective subcategory

• reflective factorization system

• Bousfield localization, Quillen reflection

References

The concept of reflective localization was originally highlighted in

• Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Springer 1967 (pdf)

A formalization in homotopy type theory of reflection onto local objects is discussed in

• Egbert Rijke, Michael Shulman, Bas Spitters, Modalities in homotopy type theory Logical Methods in Computer Science 16:1, 2020 (arXiv:1706.07526 doi)