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Mikhail Bondarko
Mikhail Bondarko
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"Almost certainly" (as far as I remember, one needs either a set of generators for $\mathcal P$ or the Vopenka principle; see Theorem 7.2.1 of Krause's https://arxiv.org/abs/0806.1324) if $\mathcal{P}$ is closed with respect to $\mathcal{T}$-coproducts then the Verdier quotient $\mathcal{T}/\mathcal{P}$ is locally small and well generated. Thus if $L:\mathcal{T} \to \mathcal{T}/\mathcal{P}$ is the localization functor then you can compose it

(i) either with the collection of functors corepresentable by the corresponding generators. This will give a collection of functors into abelian groups whose kernel is $\mathcal{T}$$\mathcal{P}$. Unfortunately, these functors do not have to respect coproducts.

(ii) or with the functor into the category $\hat{\mathcal{S}}$ of coherent functors, where $\mathcal{S}$ is the $\coprod$-closure in $\mathcal{T}/\mathcal{P}$ of a perfect set of generators $\mathcal{S}_0$; see Krause's https://eudml.org/doc/49219. You will obtain a functor that respect coproducts and whose target is AB4.

I don't know how to respect coproducts and make the target Grothendieck abelian simultaneously, sorry. Possibly, Krause's papers on localizations contain a solution.:)

"Almost certainly" (as far as I remember, one needs either a set of generators for $\mathcal P$ or the Vopenka principle; see Theorem 7.2.1 of Krause's https://arxiv.org/abs/0806.1324) if $\mathcal{P}$ is closed with respect to $\mathcal{T}$-coproducts then the Verdier quotient $\mathcal{T}/\mathcal{P}$ is locally small and well generated. Thus if $L:\mathcal{T} \to \mathcal{T}/\mathcal{P}$ is the localization functor then you can compose it

(i) either with the collection of functors corepresentable by the corresponding generators. This will give a collection of functors into abelian groups whose kernel is $\mathcal{T}$. Unfortunately, these functors do not have to respect coproducts.

(ii) or with the functor into the category $\hat{\mathcal{S}}$ of coherent functors, where $\mathcal{S}$ is the $\coprod$-closure in $\mathcal{T}/\mathcal{P}$ of a perfect set of generators $\mathcal{S}_0$; see Krause's https://eudml.org/doc/49219. You will obtain a functor that respect coproducts and whose target is AB4.

I don't know how to respect coproducts and make the target Grothendieck abelian simultaneously, sorry. Possibly, Krause's papers on localizations contain a solution.:)

"Almost certainly" (as far as I remember, one needs either a set of generators for $\mathcal P$ or the Vopenka principle; see Theorem 7.2.1 of Krause's https://arxiv.org/abs/0806.1324) if $\mathcal{P}$ is closed with respect to $\mathcal{T}$-coproducts then the Verdier quotient $\mathcal{T}/\mathcal{P}$ is locally small and well generated. Thus if $L:\mathcal{T} \to \mathcal{T}/\mathcal{P}$ is the localization functor then you can compose it

(i) either with the collection of functors corepresentable by the corresponding generators. This will give a collection of functors into abelian groups whose kernel is $\mathcal{P}$. Unfortunately, these functors do not have to respect coproducts.

(ii) or with the functor into the category $\hat{\mathcal{S}}$ of coherent functors, where $\mathcal{S}$ is the $\coprod$-closure in $\mathcal{T}/\mathcal{P}$ of a perfect set of generators $\mathcal{S}_0$; see Krause's https://eudml.org/doc/49219. You will obtain a functor that respect coproducts and whose target is AB4.

I don't know how to respect coproducts and make the target Grothendieck abelian simultaneously, sorry. Possibly, Krause's papers on localizations contain a solution.:)

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Mikhail Bondarko
Mikhail Bondarko
  • 16.9k
  • 4
  • 34
  • 97

"Almost certainly" (as far as I remember, one needs either a set of generators for $\mathcal P$ or the Vopenka principle; see Theorem 7.2.1 of Krause's https://arxiv.org/abs/0806.1324) if $\mathcal{P}$ is closed with respect to $\mathcal{T}$-coproducts then the Verdier quotient $\mathcal{T}/\mathcal{P}$ is locally small and well generated. Thus if $L:\mathcal{T} \to \mathcal{T}/\mathcal{P}$ is the localization functor then you can compose it

(i) either with the collection of functors corepresentable by the corresponding generators. This will give a collection of functors into abelian groups whose kernel is $\mathcal{T}$. Unfortunately, these functors do not have to respect coproducts.

(ii) or with the functor into the category $\hat{\mathcal{S}}$ of coherent functors, where $\mathcal{S}$ is the $\coprod$-closure in $\mathcal{T}/\mathcal{P}$ of a perfect set of generators $\mathcal{S}_0$; see Krause's https://eudml.org/doc/49219. You will obtain a functor that respect coproducts and whose target is AB4.

I don't know how to respect coproducts and make the target Grothendieck abelian simultaneously, sorry. Possibly, Krause's papers on localizations contain a solution.:)