Timeline for Local Joyal-simplicial presheaves?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 18, 2015 at 12:04 | history | edited | Peter Arndt | CC BY-SA 3.0 |
fixed broken link
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| Nov 17, 2009 at 16:44 | vote | accept | Urs Schreiber | ||
| Nov 17, 2009 at 16:18 | comment | added | D.-C. Cisinski | What about considering simply the dendroidal nerve of the little k-cubes operad C_k (i.e. apply the discrete dendroidal nerve to Hom(U,C_k) for any manifold U): I suspect it gives a dendroidal manifold which acts on k-fold loop stacks (i.e. you should get a canonical map from C_k to X^S^k in the homotopy category of "dendroidal stacks"). | |
| Nov 17, 2009 at 16:14 | comment | added | Urs Schreiber | Oh, that's sufficient, already? Great, that removes a huge headache I had. I'll digest this a bit and then might get back to you. | |
| Nov 17, 2009 at 15:53 | comment | added | D.-C. Cisinski | Sufficient conditions on V=D so that Funct(C^op,V) has a local model structure is that V is combinatorial, and such that the generating (trivial) cofibrations have cofibrant domains (Barwick only does the case where V is furthermore symmetric monoidal with cofibrant unit, but the monoidal structure is not really needed though; Ayoub does it without monoidal structures, but he requires furthermore that V is left proper and that the cofibrations are monomorphisms). Anyway, in practice, that just works well! | |
| Nov 17, 2009 at 15:27 | comment | added | Urs Schreiber | My motivating construction is currently this: I am looking at oo-stacks on the category Diff of all manifolds. Given such a beast X, there is an obvious notion of its internal k-fold loop oo-stack X^S^k. This should have an action not just of the topological E-k-operad, but of a Diff-parameterized version of that. I.e. a "smooth E-k-action". This should exist, and I would like to get my hands on it in the form of a dendroidal presheaf on Diff. | |
| Nov 17, 2009 at 15:26 | comment | added | Urs Schreiber | Wow, that's good to know. I was distracted for a minute and still need to look at that barwick reference. So what are the general conditions on the combinatorial model category C such that the gloabl structure on Funct(D,C) has a localization with respect to a Grothendieck topology on D? | |
| Nov 17, 2009 at 14:33 | comment | added | D.-C. Cisinski | The case where V is the model structure on dendroidal sets works well too. What kind of problem do you have in mind? | |
| Nov 17, 2009 at 14:22 | comment | added | Urs Schreiber | And thirdly, since I kinddly have your attention: what I am really interested in is playing this game with your model structore on dendroidal sets, i.e. consider local dendroidal set valued presheaves. Have you thought about that? | |
| Nov 17, 2009 at 14:22 | comment | added | D.-C. Cisinski | For the existence of the left Bousfield localizations, the proof of Barwick and Ayoub consist indeed to show it is sufficient to invert a small set of maps (the difficulty is essentially the same as for the classical case). As for the comparisons between Rezk model structures and model structures for Segal higher categories, they are not written yet, unfortunately (even though it is true). | |
| Nov 17, 2009 at 14:02 | comment | added | Urs Schreiber | It is to be expected that the localization of (oo,n)-cat valued presheaves using the Rezk model structure yields the Simpson-Hirschowitz construction. But is this actually known in any detail? | |
| Nov 17, 2009 at 14:01 | comment | added | Urs Schreiber | Thanks a whole lot! But to apply general results on existence of Bousfield localization, don't we need that the homotopy classes of local weak equivalences forms a small set? For the local model structure on simplicial presheaves Joyal-Jardine's construction may be regarded as proving that the localization exists event though there is not a small set of local weak equivalences. Is the generalization of this actually in the literature? (I'll have a look at the Barwick reference, it seems I am aware of a similar but different document). | |
| Nov 17, 2009 at 13:52 | history | answered | D.-C. Cisinski | CC BY-SA 2.5 |