Timeline for Characterizing positivity of formal group laws
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 4, 2018 at 22:43 | comment | added | Pedro | @JairTaylor I'm saying that axiom follows from the others you have. Regarding the operad business, as soon as I have something concrete I will let you know, but I think it is definitely doable and all the hard work and the crucial step is already contained in the Claim inside your proof of Theorem 2.1. The translation should not be complicated. | |
| Feb 4, 2018 at 22:24 | comment | added | Jair Taylor | @PedroTamaroff If you think you know the right way to re-write the definitions in terms of operads and prove Theorem 2.1 using this language I'd be interested to see it. | |
| Feb 4, 2018 at 22:23 | comment | added | Jair Taylor | @PedroTamaroff Are you saying we can replace axiom (iii) with something else? It seems like you are using some form of transitivity for contraction. The original definition is probably not optimal. | |
| Feb 4, 2018 at 21:04 | comment | added | Pedro | As a further comment, @JairTaylor, it seems the axiom that contraction preserves edge sets (which one could call "leaf sets" to be in tune with the operadic intuition) is redundant. If we write $x(T\to i)$ for the contraction of a set $T$ to $i$ in a structure $x$, and $x_T$ for the restriction of $x$ to $T$, then for any $T'\subseteq T$, we always have that $(x_T)_{T'} = x_{T'}$ (some natural transitivity of restriction), and that $x(T'\to i) = x(T\to i)\circ_i x_T(T'\to i)$, so $T(T'\to i)$, the set obtained by contracting $T'$ to a point in $T$, is an edge set of $x(T'\to i)$, as required. | |
| Feb 4, 2018 at 21:01 | comment | added | Pedro | (The catch here is to understand how $X^M$ and $X^C$ naturally become shuffle operads, and show $X^M$ becomes a right $X^C$-module. This seems to depend on understanding how minimal edge sets and connected components of hypergraphs change upon the operadic composition one can obviously endow $X$ with, using the operations defined in the thesis.) | |
| Feb 4, 2018 at 20:58 | comment | added | Pedro | @JairTaylor and Neil: From every contractible species $X$, you should be able to produce two shuffle operads (I haven't realized how to define composition here), one $X^C$, depending on connected components of hypergraphs of structures, and another $X^M$ depending on minimal edge sets on hypergraphs of structures, and it seems you main theorem rests on the fact that $X^M$ is a free right $X^C$-module over $X$. In fact, the arrows $A_n\to B_n$ for $n\in\mathbb N$ assemble to give a weighted bijection $X\circ X^C \to X^M$, and what remains to verify is this is a map of right $X^C$-modules. | |
| Feb 2, 2018 at 16:30 | comment | added | Tom Copeland | Perhaps you would be interested in Petersen's answer to mathoverflow.net/questions/259374/…. | |
| Feb 1, 2018 at 17:01 | comment | added | Jair Taylor | Yes, you can formulate it this way. Actually, noticing that experimentally was what originally lead me to this conjecture. I do think contractible species are related to operads, but I wasn't aware of them when I wrote this. | |
| Feb 1, 2018 at 11:13 | history | answered | Neil Strickland | CC BY-SA 3.0 |