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 beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry 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-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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 stack 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
    • CommentTimeNov 23rd 2009

    saw activity at simple object and started a tiny section with examples.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 29th 2019

    Looking at this article: what is the need for having a zero object as opposed to just a terminal? In other words, would it be worse to say an object is simple if it has precisely two quotients, itself and 11?

    I’m particularly interested in the case of rings and commutative rings. One definition of simple ring is that it has precisely two two-sided ideals, and this condition is equivalent to the one I’m proposing. Similarly, a simple commutative ring under my proposal is just a field.

    (It may be that the concept becomes interesting only under additional assumptions, such as Barr-exactness or something in that vein. That’s why I phrased my question as I did: would my proposal be any worse than the one given?)

    • CommentRowNumber3.
    • CommentAuthorAlizter
    • CommentTimeJul 31st 2019

    The definition of simple ring on the nlab is RR is simple as a bimodule. This seems to be equivalent to your definition. As far as I can tell your definition should subsume that definition of simple ring. I think the terminology becomes problematic for Lie algebras however.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 31st 2019

    Yeah, for Lie algebras there are practical reasons for excluding the 1-dimensional case.

    But I’m asking a more general theoretical question (insofar as there should be a general “theory” of simple objects in categories).

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)