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nLab > Latest Changes: additive functor

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2012
    • (edited Oct 4th 2012)

    at additive functor there was a typo in the diagram that shows the preservation of biproducts. I have fixed it.

    Also formatted a bit more.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2012

    Added a pointer to an old article by Watt and stated the theorem that a right exact additive functor between categories of modules comes from tensoring with a bimodule.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeOct 4th 2012
    • (edited Oct 4th 2012)

    It is Charles E. Watts, rather than Watt. Eilenberg had independently the same result.

    I have long time ago created an entry Eilenberg-Watts theorem about the right exact functors between the categories of modules over rings. Maybe this material you added on right exact functors between categories of modules should be integrated there instead. I think this is rather specific material which is not belonging really into a general entry on additive functors as the result is specific about module categories, though a short line with a remark and link to the specialized entry Eilenberg-Watts theorem makes sense. A good reference for this material are intro chapters in Bass’s book on algebraic K-theory. There are some generalizations (and dualizations) for some other setups like in some works of Pareigis, then Schauenburg and so on, in my memory.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2012

    It is Charles E. Watts, rather than Watt.

    Thanks; I have fixed it. I remember thinking “Haha, it’s not drummer Charlie Watts ” …

    I have long time ago created an entry Eilenberg-Watts theorem

    Ah, thanks, I have linked to it now. (I expect there are many more entries of yours which I will never find because you don’t link to them…)

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeOct 4th 2012

    But it is linked, e.g. at the main entry abelian category.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2012
    • (edited Oct 4th 2012)

    But it is a statement about

    1. additive functors

    2. and exact functors

    3. between categories of modules.

    So to guarantee that it isn’t duplicated when somebody else comes from any of these three ingredient-entries, it should be linked to from

    1. additive functor

    2. exact functor

    3. category of modules

    (I have done that meanwhile.)

  7. added solid abelian group as an example of an additive functor

    Anonymous

    diff, v16, current

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeApr 14th 2023

    Is it really true that Hom is additive as a bifunctor? It is obviously additive in each variable, but I can’t see how it preserves direct sums in the product category A op×AA^{op} \times A.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2023
    • (edited Apr 15th 2023)

    Right, it’s not.

    I have fixed it (here) by deleting the word “and” in “and in both arguments separately”.

    Also added a half-sentence pointing to the fact that hom-functors preserve limits in each variable separately.

    diff, v17, current

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