Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration-theory internal-categories k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 23rd 2010

    Wrote a section on the associated monad at operad, in terms of the framework introduced under the section titled Preparation.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 23rd 2010

    @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.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 23rd 2010

    Anyone else have an opinion about Harry’s suggestion? As for myself, I’ll think about it and maybe sleep on it.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMay 23rd 2010

    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.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 24th 2010
    • (edited May 24th 2010)

    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 opSet^{\mathbb{P}^{op}} is the free symmetric monoidally cocomplete category on one generator.

    (2) Given symmetric monoidally cocomplete CC, DD, let SymMonCoc(C,D)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 opSymMonCoc(Set op,Set op)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 opSet^{\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.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2010

    That’s a good idea, Todd.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeFeb 18th 2011

    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.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 19th 2011

    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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)