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In general, given a category and a class of morphisms, one may ask for the localization , or more specifically for the reflective subcategory of -local objects (the reflective localization). Similarly for variants in higher category, such as localization of an (∞,1)-category.
If has finite products, then for a given object , one may take to be the class of morphisms of the form
where is any object, and where is the terminal object, and where denotes the Cartesian product functor.
The reflective localization at such a class of morphisms is often referred to as homotopy localization at the object or similar.
The idea is that if is, or is regarded as, an interval object, then “geometric” left homotopies between morphisms are, or would be, given by morphisms out of , and hence forcing the projections to be equivalences means forcing all morphisms to be homotopy invariant with respect to .
Typically this is considered in the case that is a locally presentable category with a small set of generating objects such that it becomes sufficient to enforce the localization only on the resulting small set of morphisms of the form .
The localization of the (∞,1)-category of (∞,1)-sheaves on the Nisnevich site at the affine line is known as A1-homotopy theory.
The localization of smooth ∞-groupoids at the real line is equivalently (geometrically discrete) ∞-groupoids.
After realizing smooth ∞-groupoids as the (∞,1)-sheaves over CartSp, , this follows from the following Prop. .
(homotopy localization at over the site of s)
Let be any site (this Def.), and write for its local projective model category of simplicial presheaves (this Prop.).
Assume that contains an object , such that every other object is a finite product , for some . (In other words, assume that is also the syntactic category of Lawvere theory.)
Consider the -homotopy localization (this Def.) of the (∞,1)-sheaf (∞,1)-topos over (this Prop.)
hence the left Bousfield localization of model categories
at the set of morphisms
(according to this Prop.).
Then this is equivalent (this Def.) to ∞Grpd (this Example),
in that the (constant functor limit)-adjunction (this Def.)
is a Quillen equivalence (this Def.).
First to see that (1) is a Quillen adjunction: Since we have a simplicial Quillen adjunction before localization
(by this Example) and since both model categories here are left proper simplicial model categories (by this Prop. and this Prop.), and since left Bousfield localization does not change the class of cofibrations (this Def.) it is sufficient to show that preserves fibrant objects (by this Prop.).
But by assumption has a terminal object , which is hence the initial object of , so that the limit operation is given just by evaluation on that object:
Hence it is sufficient to see that an injectively fibrant simplicial presheaf is objectwise a Kan complex. This is indeed the case, by this Prop..
To check that (1) is actually a Quillen equivalence, we check that the derived adjunction unit and derived adjunction counit are weak equivalences:
For any simplicial set (necessarily cofibrant), the derived adjunction unit is
where is a fibrant replacement (this Def.). But is clearly the identity functor and the plain adjunction unit is the identity morphism, so that this composite is just itself, which is indeed a weak equivalence.
For the other case, let be fibrant. This means (by this Prop.) that is fibrant in the injective model structure on simplicial presheaves as well as in the local model structure, and is a derived--local object (this Def.), in that the derived hom-functor out of any into is a weak homotopy equivalence:
But since is fibrant, this derived hom is equivalent to the ordinary hom-functor (this Lemma), and hence with the Yoneda lemma (this Prop.) we have that
is a weak equivalence, for all . By induction on this means that in fact
is a weak equivalence for all . But these are just the components of the adjunction counit
which is hence also a weak equivalence. Hence for the derived adjunction counit
to be a weak equivalence, it is now sufficient to see that the value of a cofibrant replacement on is a weak equivalence. But by definition of the weak equivalences of simplicial presheaves these are objectwise weak equivalences.
For more references see also at motivic homotopy theory.
Last revised on January 10, 2019 at 18:10:12. See the history of this page for a list of all contributions to it.