Timeline for Free Symmetric Operads and $\mathbb{S}$-modules
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 10, 2021 at 15:04 | comment | added | Pedro | It is useful to note that if $X$ is binary, for example, then the free operad $F$ on $X$ has $F(2) = X(2)$, so this is a free $S_2$-module iff $X(2)$ is. It follows that you can pick counterexamples by hand. Similarly, $F(3)$ has dimension $3$, and so cannot be free as an $S_3$-module, $F(4)$ has dimension strictly less than $24$, etc: the answer follows purely by linear algebraic considerations. :) | |
| Jun 10, 2016 at 18:26 | history | edited | emmagvr | CC BY-SA 3.0 |
added 1 character in body
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| S May 27, 2016 at 12:14 | history | suggested | Samuele Giraudo | CC BY-SA 3.0 |
Some improvements.
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| May 27, 2016 at 11:53 | review | Suggested edits | |||
| S May 27, 2016 at 12:14 | |||||
| May 26, 2016 at 9:42 | vote | accept | emmagvr | ||
| May 26, 2016 at 7:38 | answer | added | Gabriel C. Drummond-Cole | timeline score: 4 | |
| May 26, 2016 at 2:37 | comment | added | Gabriel C. Drummond-Cole | What you say about colimits is not true because there are non-free maps between free S-modules. For instance, over Q, you can get the free operad on any quotient S-module of a free S-module by applying the free functor to the quotient map followed by a splitting back to the free S-module and then pushing out along the zero map. | |
| May 25, 2016 at 22:15 | comment | added | Denis Nardin | To clarify, I think that S-module in this context means what is also known as "symmetric sequences" (a.k.a. functor on finite sets and bijections) | |
| May 25, 2016 at 18:45 | comment | added | emmagvr | By "underlying S-module", the S-module obtained after apply to an operad the forgetful functor. And by "free" that the action of symmetric groups in this S-module is free. | |
| May 25, 2016 at 18:44 | review | Close votes | |||
| May 27, 2016 at 6:04 | |||||
| May 25, 2016 at 18:28 | comment | added | Ryan Budney | What do you mean by "having an underling free S-module"? | |
| May 25, 2016 at 17:01 | history | asked | emmagvr | CC BY-SA 3.0 |