Timeline for Homotopy theory for small strict semimonoidal topologically enriched categories
Current License: CC BY-SA 4.0
10 events
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| Sep 5, 2023 at 12:48 | comment | added | Simon Henry | In the case of Cisinski-Olschok theory, the requirement that every object is cofibrant fron Olschok paper can just be dropped if you are willing to work with semi-model categories (See section 6 of arxiv.org/pdf/2005.02360.pdf ) , so with this kind of methods you can chose pretty much the cofibrations you want. Though in this case I feel like this approach only works when the set of objects is fixed and one still needs to go through a similar argument to allow the set of objects to vary... | |
| Sep 5, 2023 at 12:42 | comment | added | Simon Henry | I agree not every objects should be cofibrant . But that doesn't have to be the case. In arxiv.org/abs/1201.1575 they chose to take an "injective" model structure in the case the set of objects is fixed (that is cofibration are the cofibration in V) but I believe they could have taken instead a projective one (where the fibrations are fibrations in V) and then cofibrant objects are "free on graphs" in a cellular sense. And in any case in the monoidal case I expect cofibrant objects should have free monoids of objects. | |
| Sep 5, 2023 at 4:58 | comment | added | Philippe Gaucher | @SimonHenry The problem I see with the two approaches is that all objects would be cofibrant. And what I think is that, as a reparametrization category, the terminal category should not be cofibrant. | |
| Sep 4, 2023 at 20:27 | comment | added | Simon Henry | If that doesn't pan out, my second approach would be to try to adapt the methods from arxiv.org/abs/1201.1575 which deals with the non-monoidal case. | |
| Sep 4, 2023 at 20:25 | comment | added | Simon Henry | I suspect there are some interesting model structure (or maybe semi-model structure), notably at least one that capture the Dwyer-Kan equivalence, but I'm not aware of any literature on this, and I don't see any quick way to obtain it. Meaning some, probably not difficult but tedious, work need to be done to build it. The only kind of shortcut I can think of is to see if the category you are looking at is enriched over Space and if so see if it can be exploited through some flavor of Cisinski-Olschok theory... | |
| Sep 4, 2023 at 17:49 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| Sep 4, 2023 at 17:49 | comment | added | Philippe Gaucher | @SimonHenry Exactly. I add in the question a reference to give some context. | |
| Sep 4, 2023 at 15:44 | comment | added | Simon Henry | Semi-monoidal means justs an associative tensor products and no units? | |
| Sep 4, 2023 at 15:32 | history | edited | Philippe Gaucher | CC BY-SA 4.0 |
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| Apr 6, 2023 at 11:56 | history | asked | Philippe Gaucher | CC BY-SA 4.0 |