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David White
David White
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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. Hirschornn'sHirschhorn'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 $$ f^*\colon \mathrm{Hom}_{\mathbf{H}_s(S)}(W,X))\to \mathrm{Hom} _{\mathbf{H}_s(S)} (V,X) $$$$ f^*\colon \mathrm{Hom}_{\mathbf{H}_s(S)}(W,X)\to \mathrm{Hom} _{\mathbf{H}_s(S)} (V,X) $$ 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?

SugestionSuggestion: 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.

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. Hirschornn'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?

Sugestion: 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.

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.

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Tintin
Tintin
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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. Hirschornn'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?

Sugestion: 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.