Timeline for Is the free algebra over an operad an algebra over that operad?
Current License: CC BY-SA 4.0
7 events
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| Jul 16, 2020 at 18:41 | comment | added | Javi | @AgustíRoig thank you for the references | |
| Jul 16, 2020 at 18:41 | comment | added | Javi | @DylanWilson you're right I was confusing the role of $V$ on each case, thanks for pointing out. | |
| Jul 16, 2020 at 18:21 | comment | added | Agustí Roig | Take any book about operads. For instance, in Fresse's "Homotopy of Operads...", you can find how the free operad algebra is an algebra over that operad on page 39. In Markl-Shnider-Stasheff's "Operads in Algebra...", it's on page 48. | |
| Jul 16, 2020 at 17:12 | comment | added | Andreas Blass | It's also useful to remember that you have the operad structure of $P$ available to work with. When you want to apply some $p\in P(r)$ to $y_1\otimes\cdots\otimes y_r$ where each $y_i$ has the form $[q_i\otimes$ some product of elements of $V]$, you'll want to combine $p$ with the $q_i$'s using the operad structure of $P$. | |
| Jul 16, 2020 at 17:04 | comment | added | Dylan Wilson | Is it possible you are conflating the two different "V"s? If V is a module, then the free thing P(V) will be a P-algebra so there is a map $P \to \mathrm{End}_{P(V)}$ not a map to $\mathrm{End}_{V}$. Maybe an additional confusion is that, if $V$ has the structure of a P-algebra, then there is a map of $P$-algebras $P(V) \to V$. | |
| Jul 16, 2020 at 16:50 | history | edited | Javi | CC BY-SA 4.0 |
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| Jul 16, 2020 at 16:15 | history | asked | Javi | CC BY-SA 4.0 |