Timeline for What is the relation between cobar duality and Feynman transform
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2017 at 9:16 | comment | added | Hao Yu | Thanks all! I got the answer,see arxiv.org/abs/1610.09439 prop.3.3 | |
| Apr 8, 2017 at 21:53 | comment | added | Dan Petersen | @Gabriel I don't think you should delete this. As far as I can tell you answered OP's original question in the body of the answer and his actual question in the comments? | |
| Apr 8, 2017 at 16:52 | comment | added | Gabriel C. Drummond-Cole | Sections 4.5-4.7 of arxiv.org/abs/1208.5543 have a detailed discussion of these signs and shifts and issues. | |
| Apr 8, 2017 at 16:51 | comment | added | Gabriel C. Drummond-Cole | Since the dg dual is just a twist of the cobar dual, if you know the answer to your question for cobar then you automatically know the answer for the dg dual, just insert the necessary shift. | |
| Apr 8, 2017 at 16:45 | comment | added | Gabriel C. Drummond-Cole | That these two things need some kind of shift can be seen easily by just looking at degrees of the underlying modules. If you look at 3.2.8 in Koszul duality for Operads, you see that the underlying complex of $BO(n)$ is $F(O^*[-1])(n)$, that is, there is a shift in the generators. On the other hand, according to the first sentences of (5.1) in Getzler–Kapranov, the underlying stable $\mathbb{S}$-module is $\mathbb{M}_{\mathfrak{D}^\vee}(\mathcal{A}^*)$ which has no shift on the operad $\mathcal{A}$. | |
| Apr 8, 2017 at 16:43 | comment | added | Gabriel C. Drummond-Cole | I don't see a precise definition for the shift in 5.9 but a shift is definitely necessary given your conventions. The Feynman transform $FP$ is not a modular operad but a $\mathfrak{K}$-twisted modular operad and the forgetful functor $Cyc$, which Getzler–Kapranov use to forget to cyclic $\mathbb{S}$-modules, NOT cyclic operads, cannot be naively lifted to cyclic operads because the signs of the composition are wrong. (Continued) | |
| Apr 8, 2017 at 16:26 | comment | added | Hao Yu | dg duality is in the paper " Koszul dyality for operads " . it is defined just after the definition of cobar duality. Again, is 5.9 right? Can you elaborate more on 5.9 because i dont quite understand what it means. I cant see why it needs a shift. Yes, I am in dg category. | |
| Apr 8, 2017 at 16:17 | comment | added | Gabriel C. Drummond-Cole | It's true that 5.9 is elliptical. Ward's theorem works just fine in the dg category but I don't know what you mean by "dg duality." If you want the details about shifts and signs then you need to be clearer about what conventions you are using. | |
| Apr 8, 2017 at 16:09 | comment | added | Hao Yu | Thanks!actually I konw statment 5.9, but that is different from what I posed here, it has an additional suspension applying to BO, right? Is what I posed here wrong? I dont quite understand why 5.9 needs a suspension operation.I have edited the question and want to know the relation for dg duality as well.Can you answer this question as well? | |
| Apr 8, 2017 at 15:25 | history | answered | Gabriel C. Drummond-Cole | CC BY-SA 3.0 |