Timeline for Monoidal Model Categories with Suspension Functor
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Oct 30, 2012 at 3:39 | comment | added | David White | As for your question about cohomology theories, the stable model structure on $Sp(M,G)$ chooses the weak equivalences to be maps which induce isomorphisms on all cohomology theories (such maps are called stable equivalences, as usual). Cohomology theories here are the appropriate analogue of $\Omega$-spectra, i.e. levelwise fibrant and the adjoint to the structure map is a weak equivalence for all $n$. | |
| Oct 30, 2012 at 3:37 | comment | added | David White | Hi Jon. It's just like a simplicial model category, but with a model category $D$ instead of $sSet$. So you need $M$ to be enriched, tensored, and cotensored over $D$, and that this is compatible, i.e. the pushout product axiom is satisfied. That axiom is also how you define a monoidal model category. BTW Hovey's paper can be found here: math.uiuc.edu/K-theory/0402/stable-model.pdf and another good reference for this question in the $\infty$ category setting is here: mathoverflow.net/questions/16224/…. | |
| Oct 30, 2012 at 1:11 | comment | added | Jonathan Beardsley | Oh not to mention, is there any way to know whether or not your stabilized objects give some kind of "cohomology "theories" on your original category? | |
| Oct 30, 2012 at 1:02 | comment | added | Jonathan Beardsley | Man David this is great, thanks! There's a lot to digest, but it sounds like it's pretty much exactly what I'm looking for. In fact, I'll probably need a cellular hypothesis in anything I'm thinking about anyway, so that's fine. When you say that $M$ is a $D$-model category, what does that mean? Somehow tensored over $D$? | |
| Oct 30, 2012 at 0:58 | vote | accept | Jonathan Beardsley | ||
| Oct 30, 2012 at 0:32 | history | answered | David White | CC BY-SA 3.0 |