Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group 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 integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages 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 stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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).

nLab > Latest Changes: semi-abelian category

Bottom of Page

1 to 4 of 4

  1.  
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 13th 2020

    Moving discussion here and summarizing content in the text

    +– {: .query} Mike: Why only rings without units (that is, rngs)? Intuitively, what important properties do the above listed examples share that are not shared by rings with units?

    Zoran Skoda: I want to know the answer as well. It might be something in the self-dual axioms. For unital rings artinian implies noetherian but not other way around; though the definitions of the two notions are dual.

    Toby: The category of unital rings and unitary ring homomorphisms has no zero object.

    Mike: Ah, right. Is it protomodular? I think I will understand this definition better from some non-examples that violate each clause individually.

    walt: It is protomodular. This follows from the main theorem of Characterization of Protomodular Varieties of Universal Algebra by Bourn and Janelidze. By that theorem any variety that contains a group will be protomodular. Unital rings only fail to be semiabelian for the trivial reason that ideals aren’t subrings.

    =–

    Maybe the result on protomodularity (with citation) mentioned by walt citing Bourn and Janelidze should be moved to CRing (and also Ring, if it holds for non-commutative rings).

    diff, v34, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2021

    Touched the wording and formatting of the Idea-section, in an attempt to streamline a little.

    diff, v35, current

  3. removing old query box:

    +– {: .query} Urs: how can I understand that this (has to?) involve the opposite category?

    Mike: Well, as the previous example shows, Set *Set_* itself is not semi-abelian. The way I’m thinking of it is that a surjection of pointed sets is not determined by its kernel, but an injection of pointed sets is determined by its cokernel. =–

    Anonymous

    diff, v37, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 25th 2023

1 to 4 of 4

  1.