nLab
reflective localization

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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 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 SS or morphism are understood as the default concept of localization, in fact often reflection onto the full subcategory of SS-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 SS-local objects satisfy the universal property of an SS-localization (only) for all left adjoint functors that invert the class SS (Prop. below).

Definition

Reflective localization

Definition

(category with weak equivalences)

A category with weak equivalences is

  1. a category 𝒞\mathcal{C},

  2. a subcategory W𝒞W \subset \mathcal{C}

such that the morphisms in WW

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

  2. satisfy two-out-of-three:

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

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

(localization of a category)

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

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

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

such that

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

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

    FDFγAAAAAAA𝒞 F 𝒟 γ ρ DF 𝒞[W 1] 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 DFD F and D FD^' F are related by a unique natural isomorphism κ\kappa compatible with ρ\rho and ρ \rho^':

(1)𝒞 F 𝒟 γ ρ DF id 𝒞[W 1] κ 𝒟 id D F 𝒞[W 1]AAAA=AAAA𝒞 F 𝒟 γ ρ D F 𝒞[W 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

𝒞[W 1]AAAAAAγAA𝒞 \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 SS of morphisms are, when they exist, reflections onto the full subcategory of SS-local objects (Def. below). Often this reflection of SS-local objects is what one is more interested in than the universal property of the SS-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 𝒞\mathcal{C} be a category and let SMor 𝒞S \subset Mor_{\mathcal{C}} be a set of morphisms. Then an object X𝒞X \in \mathcal{C} is called an SS-local object if for all AsBSA \overset{s}{\to} B \; \in S the hom-functor from ss into XX yields a bijection

Hom 𝒞(s,X):Hom 𝒞(B,X)AAAAHom 𝒞(A,X), 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 AfXA \overset{f}{\longrightarrow} X extends uniquely along ss to BB:

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

We write

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

for the full subcategory of SS-local objects.

Definition

(reflection onto full subcategory of local objects)

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

𝒞 SιAALAA𝒞. \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

𝒞 LAAιAAAALAA𝒞 \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 WL 1(Isos)W \coloneqq L^{-1}(Isos) of morphisms that are sent to isomorphisms by the reflector LL.

Proof

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

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

𝒞 F 𝒟 L DF 𝒞 LAAAA𝒞 id 𝒞 F 𝒟 L η ι 𝒞 L \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 η\eta are inverted by LL, and hence by assumption they are also inverted by FF, so that on the right the natural transformation F(η)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 FD^' F via a natural isomorphism ρ\rho. Pasting this now with the adjunction counit

𝒞 F 𝒟 ι ϵ L ρ D F 𝒞 L id 𝒞 L \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 ϵρ\epsilon \cdot \rho between DFD FD F \simeq D^' F. Moreover, this is compatible with F(η)F(\eta) according to (1), due to the triangle identity:

𝒞 id 𝒞 F 𝒟 id η ι ϵ L ρ D F 𝒞 L id 𝒞 LAAAA=AAAA𝒞 F 𝒟 ρ 𝒞 L \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 LL 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 W𝒞W \subset \mathcal{C} be a category with weak equivalences (Def. ). If its reflective localization (Def. ) exists

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

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

Proof

We need to show that

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

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

The first statement follows directly with the adjunction isomorphism:

Hom 𝒞(w,ι(X))Hom 𝒞[W 1](L(w),X) 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 YY is WW-local. Observe that then YY is also local with respect to the class

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

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

Since by idempotency the adjunction unit η Y\eta_Y is in W satW_{sat}, this implies that we have a bijection of the form

Hom 𝒞(η Y,Y):Hom 𝒞(ιL(Y),Y)Hom 𝒞(Y,Y). 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 Yid_Y has a preimage η Y 1\eta_Y^{-1} under this function, hence a left inverse to η\eta:

η Y 1η Y=id Y. \eta_Y^{-1} \circ \eta_Y \;=\; id_Y \,.

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

Hom 𝒞(η Y 1,ιL(Y)):Hom 𝒞(Y,ιL(Y))Hom 𝒞(ιL(Y),ιL(Y)), 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 η Y 1\eta_Y^{-1} has a left inverse (η Y 1) 1η Y 1(\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 η Y 1\eta^{-1}_Y is an inverse morphism to η Y\eta_Y, hence that η Y:YιL(Y)\eta_Y \;\colon\; Y \to \iota L (Y) is an isomorphism and hence that YY is isomorphic to an object in 𝒞[W 1]ι𝒞\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}.

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

\,

Proposition

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

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

Proof

Write

𝒞 SAAιAAAALAA𝒞 \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 SS-local objects.

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

Now let

(FG):𝒞GAAFAA𝒟 (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 FF inverts the morphisms in SS. By the adjunction hom-isomorphism it follows that GG takes values in SS-local objects. This in turn implies, now via the Yoneda embedding for 𝒟\mathcal{D}, that FF inverts all SS-local morphisms, and hence all morphisms that are inverted by LL.

Thus the essentially unique factorization of FF through LL now follows by Prop. .

References

The concept of refleftive localization was originally highligted in

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

Last revised on March 13, 2019 at 02:30:12. See the history of this page for a list of all contributions to it.