Timeline for Equivariant model structure on $G-\mathrm{Gpd}$
Current License: CC BY-SA 3.0
9 events
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| Feb 13, 2015 at 9:39 | comment | added | user2664 | @Denis:yes cofibrations are not required to be injective on arrow but trivial cofibrations are since they are injective on objects and fully faithful. Do you think there is some hope that the nerve functor maps trivial cofibrations of groupoids to trivial cofibrations of simplicial sets ? | |
| Feb 13, 2015 at 9:37 | comment | added | user2664 | @Dmitri: yes, of course for limits! Thanks. | |
| Feb 13, 2015 at 9:09 | comment | added | Dmitri Pavlov | The model structure on groupoids is transferred from simplicial sets via the nerve functor, which by definition preserves fibrations and weak equivalences. Being a right adjoint it automatically preserves all limits. See Section 2 of Hollander's “A homotopy theory for stacks”, for example. | |
| S Feb 13, 2015 at 5:43 | history | suggested | No Name | CC BY-SA 3.0 |
Improved readability
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| Feb 13, 2015 at 5:20 | review | Suggested edits | |||
| S Feb 13, 2015 at 5:43 | |||||
| Feb 12, 2015 at 23:17 | comment | added | Denis Nardin | I am sure that the nerve preserves weak equivalences (since it sends them to homotopy equivalences). I checked the "canonical model structure" on groupoids and I discovered that the cofibrations are not what I thought they were (apparently cofibrations are not required to be injections on arrows). I am sorry I do not have references. | |
| Feb 12, 2015 at 21:08 | comment | added | user2664 | I think you mean that the nerve functor preserves fibrations and trivial fibrations, not "weak equivalences and cofibrations". Right ? Do you have some reference for the preservation of some limits/colimits by the nerve functor ? | |
| Feb 12, 2015 at 17:59 | comment | added | Denis Nardin | I don't know of a reference but there's a canonical inclusion of $Gpd$ into $sSet$ (by taking the nerve) that preserves weak equivalences and cofibrations and it looks like the colimits you want to study are preserved by this inclusion, so you should be able to backport the result from the Quillen model structure | |
| Feb 12, 2015 at 16:02 | history | asked | user2664 | CC BY-SA 3.0 |