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Fixed minor typos, added links, added a top level tag, reworded to be more precise about lifting property
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David White
David White
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I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.

Let $S$ be a finite dimensional noetherian schemeNoetherian scheme and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inverting $\mathbb{A}^1$-stable equivalences we obtain Voevodsky's stable homotopy category Voevodsky's stable homotopy category $\mathbf{SH}(S)$. My question is:

Is there a model structure on $\mathbf{Spt}(S)$, having $\mathbf{SH}(S)$ as homotopy category, such that every object is fibrant? If so, could you provide a reference?

For example, does the obvious candidate, given by the class of $\mathbb{A}^1$-stable equivalences as weak equivalences, surjective morphisms as fibrations, and those havingcofibrations defined via the adequateleft lifting property, definesdefine a model structure on $\mathbf{Spt}(S)$?

I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.

Let $S$ be a finite dimensional noetherian scheme and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inverting $\mathbb{A}^1$-stable equivalences we obtain Voevodsky's stable homotopy category $\mathbf{SH}(S)$. My question is:

Is there a model structure on $\mathbf{Spt}(S)$, having $\mathbf{SH}(S)$ as homotopy category, such that every object is fibrant? If so, could you provide a reference?

For example, the obvious candidate given by the class of $\mathbb{A}^1$-stable equivalences as weak equivalences, surjective morphisms as fibrations, and those having the adequate lifting property, defines a model structure on $\mathbf{Spt}(S)$?

I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.

Let $S$ be a finite dimensional Noetherian scheme and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inverting $\mathbb{A}^1$-stable equivalences we obtain Voevodsky's stable homotopy category $\mathbf{SH}(S)$. My question is:

Is there a model structure on $\mathbf{Spt}(S)$, having $\mathbf{SH}(S)$ as homotopy category, such that every object is fibrant? If so, could you provide a reference?

For example, does the obvious candidate, given by the class of $\mathbb{A}^1$-stable equivalences as weak equivalences, surjective morphisms as fibrations, and cofibrations defined via the left lifting property, define a model structure on $\mathbf{Spt}(S)$?

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Tintin
Tintin
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Injective model Model category structure on spectra

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Tintin
Tintin
  • 2.9k
  • 17
  • 33

Injective model category structure on spectra

I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.

Let $S$ be a finite dimensional noetherian scheme and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inverting $\mathbb{A}^1$-stable equivalences we obtain Voevodsky's stable homotopy category $\mathbf{SH}(S)$. My question is:

Is there a model structure on $\mathbf{Spt}(S)$, having $\mathbf{SH}(S)$ as homotopy category, such that every object is fibrant? If so, could you provide a reference?

For example, the obvious candidate given by the class of $\mathbb{A}^1$-stable equivalences as weak equivalences, surjective morphisms as fibrations, and those having the adequate lifting property, defines a model structure on $\mathbf{Spt}(S)$?