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 of morphisms are understood as the default concept of localization, in fact often reflection onto the full subcategory of -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 -local objects satisfy the universal property of an -localization (only) for all left adjoint functors that invert the class (Prop. below).
(category with weak equivalences)
A category with weak equivalences is
a category ,
such that the morphisms in
include all the isomorphisms of ,
satisfy two-out-of-three:
If for , any two composable morphisms in , two out of the set are in , then so is the third.
Let be a category with weak equivalences (Def. ). Then the localization of at is, if it exists
such that
sends all morphisms in to isomorphisms,
is universal with this property: If is any functor with this property, then it factors through , up to natural isomorphism:
and any two such factorizations and are related by a unique natural isomorphism compatible with and :
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 of morphisms are, when they exist, reflections onto the full subcategory of -local objects (Def. below). Often this reflection of -local objects is what one is more interested in than the universal property of the -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 be a category and let be a set of morphisms. Then an object is called an -local object if for all the hom-functor from into yields a bijection
hence if every morphism extends uniquely along to :
We write
for the full subcategory of -local objects.
(reflection onto full subcategory of local objects)
Let be a category and set be a sub-class of its morphisms. Then the reflection onto local -objects (often called “localization at the collection ”) is, if it exists, a left adjoint to the full subcategory-inclusion of the -local objects (2):
(reflective subcategories are localizations)
Every reflective subcategory-inclusion
is the reflective localization at the class of morphisms that are sent to isomorphisms by the reflector .
Let be a functor which inverts morphisms that are inverted by .
First we need to show that it factors through , up to natural isomorphism. But consider the following whiskering of the adjunction unit with :
By idempotency, the components of the adjunction unit are inverted by , and hence by assumption they are also inverted by , so that on the right the natural transformation is indeed a natural isomorphism.
It remains to show that this factorization is unique up to unique natural isomorphism. So consider any other factorization via a natural isomorphism . Pasting this now with the adjunction counit
exhibits a natural isomorphism between . Moreover, this is compatible with according to (1), due to the triangle identity:
Finally, since 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 be a category with weak equivalences (Def. ). If its reflective localization (Def. ) exists
then is equivalently the inclusion of the full subcategory on the -local objects (Def. ), and hence is equivalently reflection onto the -local objects, according to Def. .
We need to show that
every is -local,
every is -local precisely if it is isomorphic to an object in .
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 is -local. Observe that then is also local with respect to the class
of all morphisms that are inverted by (the “saturated class of morphisms”): For consider the hom-functor to the opposite of the category of sets. But assumption on this takes elements in to isomorphisms. Hence, by the defining universal property of the localization-functor , it factors through , up to natural isomorphism.
Since by idempotency the adjunction unit is in , this implies that we have a bijection of the form
In particular the identity morphism has a preimage under this function, hence a left inverse to :
But by 2-out-of-3 this implies that . Since the first item above shows that is -local, this allows to apply this same kind of argument again,
to deduce that also has a left inverse . But since a left inverse that itself has a left inverse is in fact an inverse morphisms (this Lemma), this means that is an inverse morphism to , hence that is an isomorphism and hence that is isomorphic to an object in .
Conversely, if there is an isomorphism from to a morphism in the image of hence, by the first item, to a -local object, it follows immediatly that also is -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 be a category and let be a class of morphisms in . Then the reflection onto the -local objects (Def. ) satisfies, if it exists, the universal property of a localization of categories (Def. ) with respect to left adjoint functors inverting .
Write
for the reflective subcategory-inclusion of the -local objects.
Say that a morphism in is an -local morphism if for every -local object the hom-functor from to yields a bijection . Notice that, by the Yoneda embedding for , the -local morphisms are precisely the morphisms that are taken to isomorphisms by the reflector .
Now let
be a pair of adjoint functors, such that the left adjoint inverts the morphisms in . By the adjunction hom-isomorphism it follows that takes values in -local objects. This in turn implies, now via the Yoneda embedding for , that inverts all -local morphisms, and hence all morphisms that are inverted by .
Thus the essentially unique factorization of through 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.