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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor 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 homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics 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.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2011

    created a simple entry ring object, just for completeness

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 20th 2015
    • (edited Aug 20th 2015)

    In the article ring object, I actually don’t know what ring operad (suitable for defining rings in general symmetric monoidal categories) is meant. The usual obstacle in defining rings in such general contexts is the distributive law a(b+c)=ab+aca(b + c) = a b + a c, in which the variable aa is repeated on one side of the equation; such repetitions need diagonal maps or something similar, which are not available in general symmetric monoidal categories MM. (There are various dodges, such as passing to cocommutative comonoids in MM, but let’s put that aside.)

    Looking at ring operad, I think something got lost in translation. In that article, there is mention of -rings that come up in e.g. stable homotopy theory, but these are just ordinary monoids with respect to smash products of spectra or of S-modules (in one way of setting things up), where S-modules are already analogous to abelian groups or \mathbb{Z}-modules. So there the “ring operad” seems to be just the vanilla monoid operad, but within a conceptual context where we are interpreting the monoid operad in an already sufficiently rich abelian-group-like context so that it makes sense to call their algebras in that context “rings”.

    Anyway, unless something is being said that I’m not understanding, I think the sentence at ring object mentioning a “ring operad” needs amendment.

    Oh, by the way: I added to ring object the description in terms of Lawvere theories.

    • CommentRowNumber3.
    • CommentAuthorZhen Lin
    • CommentTimeAug 20th 2015

    The phrasing of the sentence is strange. One should speak of algebras over an operad, not modules. And the only way I know of defining rings as algebras over an operad is as algebras in Ab\mathbf{Ab} over the associative operad. One could also define commutative rings as algebras over the commutative operad.

    I guess the point is that for spectra, there is a whole sequence of operads between the associative operad and the commutative operad. Maybe the “anonymous coward” can explain?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 20th 2015

    I think I’ve heard “module over an operad” more and more. I don’t think it is intrinsically bad terminology, but I prefer the traditional “algebra” mainly because there is little chance of misunderstanding; meanwhile “module” has another meaning in operad theory (a module over an algebra over an operad).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2015
    • (edited Aug 20th 2015)

    @Zhen Lin, I doubt that it helps to wait for the anonymous coward to come by and explain. Just go ahead and improve the passage, given that you understand the matter. If you do feel you need to cross-check with somebody who has a real chance to show up here, try Zoran (who added the sentence in question in rev#2) or Toby (who added the pointer to “ring operad” in rev #4).

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 20th 2015

    So there the “ring operad” seems to be just the vanilla monoid operad, but within a conceptual context where we are interpreting the monoid operad in an already sufficiently rich abelian-group-like context so that it makes sense to call their algebras in that context “rings”.

    I think this is fairly common in stable homotopy theory.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 20th 2015

    Re #6: right. But (as you probably realize) my point was that to the best of my knowledge there is no “ring operad” for use in general symmetric monoidal categories, referring to a statement at ring object, and that the pointer to ring operad there doesn’t explain it away.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeAug 20th 2015

    Right.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 20th 2015

    Eh, well, I tried to cobble something together at ring object. Experts, please feel free to amend whatever got said in the Idea section.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeAug 21st 2015

    Looks good to me!

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2015

    Thanks. I have added a bunch of hyperlinks to the keywords at ring object and I have edited ring operad to at least serve as a redirect to the actual entries that we have on the topic.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 21st 2015

    Well, thanks to you too, Urs.

    There’s still the matter of the page title “ring operad”, which I continue to think is not at all good (or does this phrase appear in the literature? I mean besides in a context where operad could mean cartesian operad = Lawvere theory). If it’s in the literature, I guess we should record it as such (with pinched nose (-: ), but perhaps with some disclaimer. Notice that the text of ring operad so far doesn’t even mention the word ’operad’.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeAug 21st 2015

    I’ve never seen it.

    • CommentRowNumber14.
    • CommentAuthorTobyBartels
    • CommentTimeAug 21st 2015

    Google gives me one legitimate hit for ‘ring operad’ outside of the nLab, which is #5 at http://math.umn.edu/~tlawson/hovey/model.html. Perhaps we could ask Mark Hovey what it means.

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)