homotopy localization


Locality and descent

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




In general, given a category 𝒞\mathcal{C} and a class WW of morphisms, one may ask for the localization 𝒞[W 1]\mathcal{C}[W^{-1}], or more specifically for the reflective subcategory of WW-local objects (the reflective localization). Similarly for variants in higher category, such as localization of an (∞,1)-category.

If 𝒞\mathcal{C} has finite products, then for a given object 𝔸𝒞\mathbb{A} \in \mathcal{C}, one may take WW 𝔸W \coloneqq W_{\mathbb{A}} to be the class of morphisms of the form

X×(𝔸!*):X×𝔸p 1X, X \times (\mathbb{A} \overset{\exists!}{\to} \ast) \;\;\colon\;\; X \times \mathbb{A} \overset{p_1}{\longrightarrow} X \,,

where XX is any object, and where *\ast is the terminal object, and where ()×():𝒞×𝒞𝒞(-) \times (-) \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C} denotes the Cartesian product functor.

The reflective localization at such a class of morphisms W 𝔸W_{\mathbb{A}} is often referred to as homotopy localization at the object 𝔸\mathbb{A} or similar.

The idea is that if 𝔸\mathbb{A} is, or is regarded as, an interval object, then left homotopies between morphisms XYX \to Y are, or would be, given by morphisms out of X×𝔸X \times \mathbb{A}, and hence forcing the projections X×𝔸XX \times \mathbb{A} \to X to be equivalences means forcing all morphisms to be homotopy invariant with respect to 𝔸\mathbb{A}.

Typically this is considered in the case that 𝒞\mathcal{C} is a locally presentable category with a small set of generating objects G iG_i such that it becomes sufficient to enforce the localization only on the resulting small set of morphisms of the form G i×(𝔸*)G_i \times (\mathbb{A} \to \ast).



(homotopy localization at 𝔸 1\mathbb{A}^1 over the site of 𝔸 n\mathbb{A}^ns)

Let 𝒞\mathcal{C} be any site (this Def.), and write [𝒞 op,sSet Qu] proj,loc[\mathcal{C}^{op}, sSet_{Qu}]_{proj, loc} for its local projective model category of simplicial presheaves (this Prop.).

Assume that 𝒞\mathcal{C} contains an object 𝔸𝒞\mathbb{A} \in \mathcal{C}, such that every other object is a finite product 𝔸 n𝔸××𝔸nfactors\mathbb{A}^n \coloneqq \underset{n \; \text{factors}}{\underbrace{\mathbb{A} \times \cdots \times \mathbb{A}}}, for some nn \in \mathbb{N}. (In other words, assume that 𝒞\mathcal{C} is also the syntactic category of Lawvere theory.)

Consider the 𝔸 1\mathbb{A}^1-homotopy localization (this Def.) of the (∞,1)-sheaf (∞,1)-topos over 𝒞\mathcal{C} (this Prop.)

Sh (𝒞) 𝔸AAιAAL 𝔸Sh (𝒞)Ho(CombModCat) Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat)

hence the left Bousfield localization of model categories

[𝒞 op,sSet Qu] proj,loc,𝔸 Qu QuAAidAAid[𝒞 op,sSet Qu] proj,locCombModCat [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc,\mathbb{A}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc} \;\; \in CombModCat

at the set of morphisms

S{𝔸 n×𝔸p 1𝔸 n} S \;\coloneqq\; \big\{ \mathbb{A}^n \times \mathbb{A} \overset{p_1}{\longrightarrow} \mathbb{A}^n \big\}

(according to this Prop.).

Then this is equivalent (this Def.) to ∞Grpd (this Example),

GrpdSh (𝒞) 𝔸AAιAAL 𝔸Sh (𝒞)Ho(CombModCat) \infty Grpd \;\simeq\; Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat)

in that the (constant functor \dashv limit)-adjunction (this Def.)

(1)[𝒞 op,sSet Qu] inj,loc,𝔸limAAconstAAsSet QuCombModCat [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu} \;\;\;\; \in CombModCat

is a Quillen equivalence (this Def.).


First to see that (1) is a Quillen adjunction: Since we have a simplicial Quillen adjunction before localization

[𝒞 op,sSet Qu] injlimAAconstAAsSet Qu [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu}

(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 lim\underset{\longleftarrow}{\lim} preserves fibrant objects (by this Prop.).

But by assumption 𝒞\mathcal{C} has a terminal object *=𝔸 0\ast = \mathbb{A}^0, which is hence the initial object of 𝒞 op\mathcal{C}^{op}, so that the limit operation is given just by evaluation on that object:

limX=X(𝔸 0). \underset{\longleftarrow}{\lim} \mathbf{X} \;=\; \mathbf{X}(\mathbb{A}^0) \,.

Hence it is sufficient to see that an injectively fibrant simplicial presheaf X\mathbf{X} 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 XsSetX \in sSet any simplicial set (necessarily cofibrant), the derived adjunction unit is

Xid Xconst(X)(𝔸 0)const(j X)(𝔸 0)const(PX)(𝔸 0) X \overset{id_X}{\longrightarrow} const(X)(\mathbb{A}^0) \overset{ const(j_X)(\mathbb{A}^0) }{\longrightarrow} const(P X)(\mathbb{A}^0)

where Xj XPXX \overset{j_X}{\longrightarrow} P X is a fibrant replacement (this Def.). But const()(𝔸 0)const(-)(\mathbb{A}^0) is clearly the identity functor and the plain adjunction unit is the identity morphism, so that this composite is just j Xj_X itself, which is indeed a weak equivalence.

For the other case, let X[𝒞 op,sSet Qu] inj,loc,𝔸 1\mathbf{X} \in [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}^1} be fibrant. This means (by this Prop.) that X\mathbf{X} is fibrant in the injective model structure on simplicial presheaves as well as in the local model structure, and is a derived-𝔸 1\mathbb{A}^1-local object (this Def.), in that the derived hom-functor out of any 𝔸 n×𝔸 1p 1𝔸 n\mathbb{A}^n \times \mathbb{A}^1 \overset{p_1}{\longrightarrow} \mathbb{A}^n into X\mathbf{X} is a weak homotopy equivalence:

Hom(p 1):Hom(𝔸 n,X)WHom(𝔸 n×𝔸 1,X) \mathbb{R}Hom( p_1 ) \;\colon\; \mathbb{R}Hom( \mathbb{A}^n , \mathbf{X}) \overset{\in W}{\longrightarrow} \mathbb{R}Hom( \mathbb{A}^n \times \mathbb{A}^1 , \mathbf{X})

But since X\mathbf{X} 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

X(p 1):X(𝔸 n)WX(𝔸 n+1) \mathbf{X}(p_1) \;\colon\; \mathbf{X}(\mathbb{A}^n) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^{n+1})

is a weak equivalence, for all nn \in \mathbb{N}. By induction on nn this means that in fact

X(𝔸 0)WX(𝔸 n) \mathbf{X}(\mathbb{A}^0) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^n)

is a weak equivalence for all nn \in \mathbb{N}. But these are just the components of the adjunction counit

const(X(𝔸 0))WϵX const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X}

which is hence also a weak equivalence. Hence for the derived adjunction counit

const(QX)(𝔸 0)const(p X(𝔸 0))const(X(𝔸 0))WϵX const (Q \mathbf{X})(\mathbb{A}^0) \overset{const(p_{\mathbf{X}}(\mathbb{A}^0))}{\longrightarrow} const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X}

to be a weak equivalence, it is now sufficient to see that the value of a cofibrant replacement p Xp_{\mathbf{X}} on 𝔸 0\mathbb{A}^0 is a weak equivalence. But by definition of the weak equivalences of simplicial presheaves these are objectwise weak equivalences.

Last revised on July 11, 2018 at 09:48:35. See the history of this page for a list of all contributions to it.