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).
(category with weak equivalences)
A category with weak equivalences is
a category $\mathcal{C}$,
a subcategory $W \subset \mathcal{C}$
such that the morphisms in $W$
include all the isomorphisms of $\mathcal{C}$,
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.
Let $W \subset \mathcal{C}$ be a category with weak equivalences (Def. ). Then the localization of $\mathcal{C}$ at $W$ is, if it exists
a category $\mathcal{C}[W^{-1}]$
a functor $\gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]$
such that
$\gamma$ sends all morphisms in $W \subset \mathcal{C}$ to isomorphisms,
$\gamma$ 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:
and any two such factorizations $D F$ and $D^' F$ are related by a unique natural isomorphism $\kappa$ compatible with $\rho$ and $\rho^'$:
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
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).
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
hence if every morphism $A \overset{f}{\longrightarrow} X$ extends uniquely along $s$ to $B$:
We write
for the full subcategory of $S$-local objects.
(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):
(reflective subcategories are localizations)
Every reflective subcategory-inclusion
is the reflective localization at the class $W \coloneqq L^{-1}(Isos)$ of morphisms that are sent to isomorphisms by the reflector $L$.
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 $F$:
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
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:
Finally, since $L$ is essentially surjective functor, by idempotency, it is clear that this is the unique such natural isomorphism.
(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
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. .
We need to show that
every $X \in \mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is $W$-local,
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:
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
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
In particular the identity morphism $id_Y$ has a preimage $\eta_Y^{-1}$ under this function, hence a left inverse to $\eta$:
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,
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.
$\,$
(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 $S$.
Write
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
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. .
The concept of reflective localization was originally highlighted in
A formalization in homotopy type theory of reflection onto local objects is discussed in
Last revised on August 31, 2024 at 20:47:25. See the history of this page for a list of all contributions to it.