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Wrote a section on the associated monad at operad, in terms of the framework introduced under the section titled Preparation.
@Todd: Perhaps we should move the detailed conceptual treatment to a separate page (including this part that you wrote just now). I was thinking of writing out some of the easy proofs of the statements, and it seems like that section is growing independently of the rest of the article. It seems like a good candidate for splitting, but I defer to your judgement.
Anyone else have an opinion about Harry’s suggestion? As for myself, I’ll think about it and maybe sleep on it.
I’m not sure why that section is called “detailed conceptual treatment” – it seems to me to be just one way of giving the definition. The previous definition is just as adequate, modulo the missing coherence diagrams. Another way of giving the definition is in terms of generalized multicategories, which is not discussed much at operad. (By the way, operad and multicategory and even globular operad were surprisingly lacking in links to generalized multicategory, so I added some.) I could see the argument for splitting it off, especially if you’re going to add detailed proofs; in line with the “zoomability” philosophy the article “operad” should maybe be more concise with a link to the details if people want them.
The section on “free operads” also looks to me as though it would fit better on a page free operad.
I think the material under “detailed conceptual treatment” could be condensed considerably just by recalling a few key facts, and then one could zoom in on these facts. Let’s see if I can do it here:
(1) $Set^{\mathbb{P}^{op}}$ is the free symmetric monoidally cocomplete category on one generator.
(2) Given symmetric monoidally cocomplete $C$, $D$, let $SymMonCoc(C, D)$ denote the category of symmetric monoidal cocontinuous functors and symmetric monoidal transformations. It follows from (1) that there is an equivalence of categories
$Set^{\mathbb{P}^{op}} \simeq SymMonCoc(Set^{\mathbb{P}^{op}}, Set^{\mathbb{P}^{op}})$(3) The right side of this equivalence is a strict monoidal category whose monoidal product is endofunctor composition. The monoidal structure transfers across the equivalence to give monoidal category structure on $Set^{\mathbb{P}^{op}}$. An operad is by definition a monoid in this monoidal category.
I don’t think that’s too cryptic a summary, and one can just “zoom in” on the terms to find out what they mean. It shouldn’t be much more than a cut-and-paste job, really, since so many of the details are currently at operad.
That’s a good idea, Todd.
Todd, do you have the source of your text listed at operad – John linked there is a pdf scan which is not of best scanning quality.
http://math.ucr.edu/home/baez/trimble/trimble_lie_operad.pdf
By the way, Loday and Valette wrote a web draft of a new book “Algebraic operads” which I now listed in operad, link is http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf.
Zoran, I don’t think I have the latex file, but I do have hard copy which I can mail you if you like. (I could probably rewrite the latex file since the paper is short, and that would give me one thing (among others) to do while I’m with my in-laws all next week, starting tomorrow.) I should probably send John the missing page 16.
I spotted one such problem in this paragraph and fixed it (the parenthetical remark there used to be enclosed in square brackets).
If there are more issues, please give some indication where to find them (e.g. name closeby strings that one can search for)
I have added publication data to:
I have added hyperlink to double functor
Do we have anywhere on the nLab a definition of operad or multicategory in terms of the binary “$\circ_i$” composition operations instead of the $n$-ary “parallel” composition operations? I don’t see it at operad or multicategory.
(This alternative definition is equivalent in the presence of all the other axioms, but when you start removing things, like the identity operations or the parallel associativity axiom, starting from the $\circ_i$ definition often gives a better result.)
On a related note, it would be nice to have a name for the two presentations of operad/multicategory. I think I’ve only seen names that involve $\circ_i$, as you write, which is not particularly convenient.
Yeah. One possibility is the phrases I also used: “binary composition” and “parallel $n$-ary composition”.
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