For a class of morphisms in a category (or higher category) , we can consider the localization functor into the category of fractions which is universal with respect to all functors inverting ; this defines its (external) saturation, the class of all morphisms which are sent to isomorphisms by , or more generally to equivalences. One also defined a collection of objects which are local with respect to these morphisms, in that these morphisms behave as isos/equivalences with respect to homming into . If has a right adjoint then it is a reflective localization, and in particular the right adjoint is fully faithful; we can abstractly consider reflective localizations of the form and start with internally saturated class of all morphisms inverted by . Then the collection of -local objects characterizes the localization. Moreover, can itself be enlarged to its external saturation, the collection of -local morphisms, that is all morphisms which are isomorphisms with respect to homing into all -local objects. In the reflective case, external and internal saturation coincide and in general external saturation is a subclass of internal saturation.
Gabriel-Zisman used the term “left-closed object for ”. This terminology is still widely used in the context of abelian categories with or without “left”; -closed is then closed in the sense of any kind of localization data (for example, being a torsion theory or a Gabriel filter). Sheaves and stacks are examples of closed objects in presheaf categories on sites/spaces.
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…
Every morphism in is -local.
There are two notions of saturation in the literature. Following Casacuberta, Frei 2000 we distinguish them as internal and external. Internal saturation of a class of morphisms is simply the class of all -local morphisms. External saturation of a class of morphisms is the class of all morphisms which are inverted in the category of fractions (localization) at class .
In both variants, the collection of morphisms is called saturated if coincides with its saturation.
is contained in its external saturation which is in turn contained in its internal saturation; the external and internal saturation coincide if the localization in has either left or right adjoint (coreflective or reflective case).
Local objects are treated, under the name of left-closed objects, in I.4.1 of
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
1-categorical picture
Carles Casacuberta, John Frei, On saturated classes of morphisms, Theory and Applications of Categories 7:4 (2000) 43–46 pdf
Carles Casacuberta, Georg Peschke, Markus Pfenniger, On orthogonal pairs in categories and localization, J. Pure & Applied Algebra 142:1 (1999) 25–33
Relation to the categorical shape theory
Last revised on August 29, 2024 at 01:37:40. See the history of this page for a list of all contributions to it.