Reflective localizations of categories and higher categories in the sense of left adjoint functors to inclusions of full subcategories (as in particular for geometric embeddings) are characterized by the collection of morphisms of which are sent by to isomorphisms, or more generally to equivalences, as well as by the collection of objects which are local with respect to these morphisms, in that these morphisms behave as equivalences with respect to homming into objects.
Let be a category and a collection of morphisms in . Then an object is -local if the hom-functor
sends morphisms in to isomorphisms in Set, i.e. if for every in , the function
is a bijection.
Conversely, a morphism is -local if for every -local object the induced morphism
is an isomorphism.
Let be an (∞,1)-category and a collection of morphisms in . Then an object is -local if the hom-functor
evaluated on induces isomorphism in the homotopy category of Top.
Conversely, a morphism is -local if for every -local object the induced morphism
induces an isomorphism in the homotopy category of Top.
Let be a model category (usefully but not necessarily a simplicial model category). And let be a collection of morphisms in .
Write for the derived hom space functor.
For instance if is a simplicial model category then this may be realized in terms of a cofibrant replacement functor and a fibrant replacement functor as
(local object, local weak equivalence)
An object is a -local object if for all in the induced morphism
is a weak equivalence (in the standard model structure on simplicial sets);
A morphism in is an -local morphism or -equivalence if for every -local object the induced morphism
is a weak equivalence.
An -localization of an object is an -local object and an -local equivalence .
An -localization of a morphism is a pair of -localizations and of objects, and a commuting square
In left proper model categories there is an equivalent stronger characterization of -locality of cofibrations .
(characterization of -local cofibrations)
Let be a left proper simplicial model category and , a collection of morphisms.
Then a cofibration is an -local weak equivalence precisely if for all fibrant -local objects the morphism
is an acyclic fibration in the standard model structure on simplicial sets.
Notice that this is stronger than the statement that is a weak equivalence not only in that it asserts in addition a fibration, but also in that it deduces this without first passing to a cofibrant replacement of and .
This is HTT, lemma A.3.7.1.
The proof makes use of the following general construction: for any morphism let be a cofibrant replacement, factor as and consider the pushout diagram
By left properness the pushout of the weak equivalence along the cofibration is again a weak equivalence and by 2-out-of-3 the morphism is a weak equivalence.
Now assume that is an -local equivalence. We need to show that is an acyclic Kan fibration for all fibrant -local . By the very definition of enriched model category it follows from being a cofibration and being fibrant that this is a Kan fibration. So it remains to show that it is a weak homotopy equivalence of simplicial sets. We know that the corresponding induced morphism
on the cofibrant replacement is a weak equivalence, by the assumption that is -local, and also, as before, a fibration, since is still a cofibration.
By homming the entire diagram above into , and using that the hom-functor sends colimits to limits, we find the pullback diagram
in SSet, which shows that is an acyclic fibration, being the pullback of an acyclic fibration.
To show that is a weak equivalence it suffices to show that all its fibers over elements are contractible Kan complexes. These fibers map to the corresponding fibers by precomposition with . By the fact that , regarded as a morphism
in the model structure on the undercategory is a weak equivalence between cofibrant objects (because is a cofibration by assumption and as being the pushout of the cofibration ) we have that precomposition with is the image under the SSet-enriched hom-functor of a weak equivalence between cofibrant objects mapping into a fibrant object
and hence, by the general properties of enriched homs between cofibrant/fibrant objects a weak equivalence. , so that indeed is contractible.
This proves the first part of the statement. For the converse statement, assume now that…
A classical textbook reference is section 3.2 of
A useful reference with direct ties to the (∞,1)-category story in the background is section A.3.7 of
Every morphism in is -local.
The collection of morphisms is called saturated if the collection of -local morphisms coincides with .
Last revised on June 29, 2018 at 12:23:50. See the history of this page for a list of all contributions to it.