Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them.

First definition: Let $\mathbf{C}$ be a simplicial model category and $\mathrm{A}$ be a set of morphism. A map $f\colon V\to W$ is an $A$-local equivalence if for any $A$-local object $X$ the map of simplicial sets $$f^*\colon \mathrm{map} (W,X)\to \mathrm{map}(V,X)$$ is a weak equivalence. Recall that $\mathrm{map}(\ , \ )$ denotes the homotopy function complex. More concretely, a simplicial set satisfying $$\pi_0(\mathrm{map} (W,X))=\mathrm{Hom}_{\mathrm{Ho}(\mathbf{C})}(W,X)$$ The definition of an $\mathrm{A}$-local object is given also in terms of $\mathrm{map}(\ , \ )$ (cf. Hirschhorn's Model Categories and their localizations 3.1.4).

Second definition: In many contexts of Voevodsky's motivic homotopy theory people use the following definition (cf Morel-Voevodsky's $\mathbb{A}^1$-homotopy theory of schemes or Riou's Categorie homotopique...). The model category is now the category of simplicial sheaves on the big (smooth) Nisnevich site with the simplical model structure (denote $\mathbf{H}_s(S)$ the homotopy category where $S$ is a scheme) or the category of spectra with the level structure . For example, MV definition would say that a map $f\colon V\to W$ is an $\mathbb{A}^1$-local equivalence if for every $\mathbb{A}^1$-local object $X$ the induced map $$f^*\colon \mathrm{Hom}_{\mathbf{H}_s(S)}(W,X)\to \mathrm{Hom} _{\mathbf{H}_s(S)} (V,X)$$ is a bijection. The definition of local objects is also given in terms of $\mathrm{Hom} _{\mathbf{H}_s(S)} (\ ,\ )$ and not $\mathrm{map}(\ , \ )$. Everyone in motivic homotopy theory states that this is a Bousfield localization.

Does anyone know how to prove this two definitions agree in motivic homotopy theory?

Suggestion: The closest to a link between these two definitions is M-V's result 2.2.8 where they state that it is equivalent for an object $X$ to be $\mathrm{A}$-local in the category of simplicial sheaves over a site and that $$f^*\colon\underline{\mathrm{Hom}} (V,X)\to \underline{\mathrm{Hom}} (W,X)$$ is a simplicial weak equivalence for any map $f$ in $\mathrm{A}$. However, note that $\underline{\mathrm{Hom}} (V,X)(S)=\mathrm{map}(V,X)$ but simplicial weak equivalences are weak equivalences at stalks, not globally.

Yes, they are the same. In order to prove it, we should show that they have the same new weak equivalences (this is enough, because both have the same cofibrations). Have a look at Barwick's paper On Left and Right Model Categories and Left and Right Bousfield Localization. In 4.45 he defines what "Enriched left Bousfield localization" means. In the case of a closed symmetric monoidal category (i.e. enriched over itself), it means you can test for new weak equivalences via the internal hom, i.e. your second definition is what he calls enriched localization. Gutierrez has considered a similar notion under the term "closed localization."

In 4.46 Barwick proves that enriched localization exists and is simply the usual localization (your first definition) with respect to $I \Box S$, where $S$ runs through the maps you want to invert, $I$ is the set of generating cofibrations, and the box means pushout product.

In your situation of interest, $I$ is obtained from the generating cofibrations in sSet, i.e. built freely from maps of the form $\partial \Delta^n \to \Delta^n$ for $n\geq 0$. In particular, every map in $S$ becomes a weak equivalence in definition 2. To show that every map in $I\Box S$ is already a weak equivalence in definition 1, look at Hirschhorn section 4.2. The way in which the model structure for definition 1 is built already includes taking the pushout product with $I$. So these maps in $I\Box S$ are trivial cofibrations in definition 1, hence weak equivalences, and we're done.

Note that we are using here that for a simplicial model category (like yours) that we can avoid framings and already have a nice formula for cosimplicial resolution. That's the real reason taking the pushout product with $I$ comes in, and why those maps in $I\Box S$ are new weak equivalences for Hirschhorn.

There are many equivalent ways of defining the local equivalences. Let $\mathcal{M}$ be a simplicial model category with a cofibrant replacement functor $Q : \mathcal{M} \to \mathcal{M}$. Then, for a fibrant object $Z$, define $$\mathbf{R}\mathrm{Hom} (-, Z) = \mathrm{Hom} (Q {-}, Z)$$ where the RHS is the simplicially enriched hom-functor. (Note that we may take $Q = \mathrm{id}$ if every object in $\mathcal{M}$ is cofibrant.) This gives a very convenient model of the homotopy function complex.

Now, fix a class $\mathcal{R}$ of fibrant objects in $\mathcal{M}$ that is closed under cotensoring, i.e. if $Z$ is a member of $\mathcal{R}$ and $K$ is a simplicial set, then there is an object $Z^K$ in $\mathcal{M}$ that is a member of $\mathcal{R}$ with an isomorphism $$\mathrm{Hom}_\mathbf{sSet} (K, \mathrm{Hom}_\mathcal{M} (-, Z)) \cong \mathrm{Hom}_\mathcal{M} (-, Z^K)$$ of simplicially enriched functors $\mathcal{M} \to \mathbf{sSet}$. For example, the class of (fibrant) $\mathcal{S}$-local objects in $\mathcal{M}$ is such a class.

Let $X \to Y$ be a morphism in $\mathcal{M}$. The following are equivalent:

• For every object $Z$ in $\mathcal{R}$, $$\mathbf{R}\mathrm{Hom} (Y, Z) \to \mathbf{R}\mathrm{Hom} (X, Z)$$ is a weak homotopy equivalence of simplicial sets.
• For every object $Z$ in $\mathcal{R}$ and every simplicial set $K$, $$\pi_0 \mathrm{Hom} (K, (\mathbf{R}\mathrm{Hom} (Y, Z)) \to \pi_0 \mathrm{Hom} (K, \mathbf{R}\mathrm{Hom} (X, Z))$$ is a bijection of sets.
• For every object $Z$ in $\mathcal{R}$, $$\pi_0 \mathbf{R}\mathrm{Hom} (Y, Z) \to \pi_0 \mathbf{R}\mathrm{Hom} (X, Z)$$ is a bijection of sets.

In particular, the $\mathcal{S}$-local equivalences can be defined entirely in terms of the homotopy category of $\mathcal{M}$. The same conclusion holds for ordinary model categories as well, but the details are more complicated because we do not have such a convenient model of the homotopy function complex.