nLab localization at geometric homotopies

Definition

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

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

$$X \times (\mathbb{A} \overset{\exists!}{\to} \ast) \;\;\colon\;\; X \times \mathbb{A} \overset{p_1}{\longrightarrow} X \,,$$

where $X$ is any object, and where $\ast$ is the terminal object, and where $(-) \times (-) \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ denotes the Cartesian product functor.

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

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

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

Examples

• The localization of the (∞,1)-category of (∞,1)-sheaves on the Nisnevich site at the affine line $\mathbb{A}^1$ is known as A1-homotopy theory.

• The localization of smooth ∞-groupoids at the real line $\mathbb{R}^1$ is equivalently (geometrically discrete) ∞-groupoids.

  After realizing smooth ∞-groupoids as the (∞,1)-sheaves over CartSp, $Smooth\infty Grpd \simeq Sh_{\infty}(CartSp)$, this follows from the following Prop. .

Proposition

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

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

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

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

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

hence the left Bousfield localization of model categories

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

at the set of morphisms

$$S \;\coloneqq\; \big\{ \mathbb{A}^n \times \mathbb{A} \overset{p_1}{\longrightarrow} \mathbb{A}^n \big\}$$

(according to this Prop.).

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

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

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

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

is a Quillen equivalence (this Def.).

Proof

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

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

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

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

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

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

$$X \overset{id_X}{\longrightarrow} const(X)(\mathbb{A}^0) \overset{ const(j_X)(\mathbb{A}^0) }{\longrightarrow} const(P X)(\mathbb{A}^0)$$

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

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

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

But since $\mathbf{X}$ 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

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

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

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

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

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

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

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

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

References

• Fabien Morel, Vladimir Voevodsky, Def. 3.1 in $\mathbb{A}^1$-homotopy theory of schemes, Publications Mathématiques de l’IHÉS, Volume 90 (1999), p. 45-143 (Numdam:PMIHES_1999__90__45_0 K-Theory:0305 )

For more references see also at motivic homotopy theory.