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  1. disambiguation page for Ore condition

    watcher

    v1, current

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeFeb 1st 2025

    But that is the same thing.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2025

    Looking at Ore set and at Ore condition in a category they look similar but not the same (even granting specialization to one-object categories).

    Also they don’t refer to each other, currently.

    Best to clarify the entries, and then we can see if to merge them or not.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2025
    • (edited Feb 2nd 2025)

    It is the same. The Ore condition is with respect to some set/class of arrows (denominator set in algebraic setup). The page Ore condition in a category has strangely covered only the case when this class is the class of all arrows (the fact that Johnstone needed this in his particular setup does not make it standard to restrict to this case). The page calculus of fractions covers general case. If the category is small and preadditive and you take one object case you get the ring case. It is a matter of taste if the cancelability condition is also included or not in (outside of ring theory) the obsolete term “Ore condition”. Making up some nonstandard terminology and giving them separate pages here confuses even those of us who spent several years on this very subject.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2025

    Ah, somehow I thought that you had created both entries. But now I see that the content at “Ore condition in a category” was started by Thomas Holder in 2016.

    Best would probably be to delete this entry and instead to boost Ore set to the point that it makes clear the generalization to categories.

    If anyone had the leisure to do that.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeFeb 3rd 2025

    Surely, I could do something about it, but more likely only after some two weeks from now, it is exams periods and so much grading at the moment. Ore condition in a category has otherwise interesting content about the role in Johnstone’s context.