Timeline for Does abstract nonsense of model categories determine the "nonlinear" morphisms of $L_\infty$ algebras?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 18, 2013 at 14:33 | comment | added | Joey Hirsh | however, the lack of a free P-algebra functor makes the connection between the ``nonlinear morphisms" between algebras over a properad and the morphisms in the homotopy category of algebras over that properad unclear (to me). | |
| Jul 18, 2013 at 14:26 | comment | added | Joey Hirsh | there's no free functor for algebras over a properad. there is however a free P-module functor, and I believe this allows for a construction which is similar to the bar construction for algebras, although the output is not a coalgebra of dual type, it is some other thing, maybe a comodule, of dual type. I believe this paper arxiv.org/abs/0807.1241 provides an example of what I'm talking about. | |
| Jul 17, 2013 at 23:49 | comment | added | Theo Johnson-Freyd | Hi @JoeyHirsh, I would accept an answer that described "the homotopy category of nonlinear morphisms" given some arbitrary replacement. I'm hoping, though, for something that would give you "nonlinear morphisms" for other (pr)operads. Recently I've been thinking a lot about the properad of Lie bialgebras (you and I talked about this a bit in March), which is Koszul, and I can compute a cofibrant replacement using this, but because there's no "free" functor for representations of a properad, I don't see how to decide what the "nonlinear morphisms" should be. | |
| Jul 17, 2013 at 23:45 | comment | added | Theo Johnson-Freyd | @DavidWhite: Yes, I've talked to Bruno here about various things. I hadn't seen that question, though! | |
| Jul 17, 2013 at 21:28 | comment | added | Joey Hirsh | Can you make this a precise math question? For example, will you be satisfied if given an arbitrary replacement, you can find the ``homotopy category of the nonlinear morphisms"? I suspect not, but it's not clear to me what exactly you want. | |
| Jul 17, 2013 at 20:39 | comment | added | Gabriel C. Drummond-Cole | This doesn't answer your question directly, but there's a picture where instead of looking at intertwining maps $V\to W$ as your morphisms, you look at maps from your operad to $End(V,W)$ viewed as a bimodule over $End(V)$ and $End(W)$. I think this gives the right picture whatever cofibrant replacement you use. | |
| Jul 17, 2013 at 20:11 | history | asked | Theo Johnson-Freyd | CC BY-SA 3.0 |