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nLab > Latest Changes: semi-abelian category

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    • 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

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