Timeline for Stabilization of a generic pointed model category
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jul 17, 2015 at 16:05 | vote | accept | Marc Nieper-Wißkirchen | ||
| Jul 14, 2015 at 20:35 | comment | added | David White | @MarcNieper-Wisskirchen: Okay, I did my best to provide an answer. I am traveling so might be very slow to respond in the comments. Hopefully the answer clarifies things a bit. | |
| Jul 14, 2015 at 20:34 | answer | added | David White | timeline score: 3 | |
| Jul 14, 2015 at 9:27 | comment | added | Dmitri Pavlov | @MarcNieper-Wißkirchen: If you're not concerned with specific examples, then one can simply take the underlying quasicategory and then stabilize it using one of the constructions in Proposition 1.4.2.24 in Higher Algebra. This is guaranteed to produce the correct answer. | |
| Jul 14, 2015 at 6:55 | comment | added | Marc Nieper-Wißkirchen | @White: Thanks. So my question remains. The question is less about a particular application than about getting a complete picture of what is going on. | |
| Jul 13, 2015 at 20:43 | comment | added | David White | @MarcNieper-Wisskirchen: Certainly not. See my recent MathOverflow question mathoverflow.net/questions/209734 Here are 3 counterexamples: the Strom model structure on Top, the absolute model structure on chain complexes, and pro model structures. | |
| Jul 13, 2015 at 17:45 | comment | added | Dmitri Pavlov | A model category is Quillen equivalent to a left proper combinatorial model category if and only if its underlying ∞-category is presentable. This is almost always satisfied in practice, though I am not sure which example you have in mind. Supplying additional context for your question would be quite useful, e.g., what kind of application do you have in mind? | |
| Jul 13, 2015 at 17:37 | comment | added | Marc Nieper-Wißkirchen | Is every model category Quillen equivalent to a left proper combinatorial one? | |
| Jul 13, 2015 at 16:40 | comment | added | Dmitri Pavlov | “Is there a canonical way to stabilize a model category without going through simplicial enrichments (and which may work with any closed model category)?”: The answer to this question, as stated, is yes: start by replacing your model category with a Quillen equivalent left proper combinatorial model category. Present the suspension as a left Quillen endofunctor (any choice works) and apply Hovey's machinery (Definition 3.3 in his paper), which does not need a simplicial enrichment. | |
| Jul 13, 2015 at 16:22 | comment | added | Dmitri Pavlov | “Does it make sense just to stabilize the homotopy category Ho(C), e.g. just to consider spectra on the homotopy level?”: It should not be very difficult to construct two model categories C and D such that Ho(C) is equivalent to Ho(D), but Ho(Stab(C)) is not equivalent to Ho(Stab(D)). | |
| Jul 13, 2015 at 11:18 | history | edited | Marc Nieper-Wißkirchen | CC BY-SA 3.0 |
reference added
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| Jul 13, 2015 at 9:48 | history | asked | Marc Nieper-Wißkirchen | CC BY-SA 3.0 |