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nLab > Latest Changes: center of an additive category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorJohn Baez
    • CommentTimeMay 15th 2023

    Showed that for any commutative ring RR, the center of the category of RR-modules is RR.

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2023

    added more hyperlinking and formatting (here)

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2023

    have reworded the Idea section (here) to make it more informative.

    (The half-sentence on unnamed “reconstruction theorems” remains as vague and mysterious as it was. If we wnt to keep this, somebody should add more information to make it worthwhile.)

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeMay 15th 2023

    Added remark: why Morita equivalent commutative rings are isomorphic.

    diff, v9, current

    • CommentRowNumber5.
    • CommentAuthorJohn Baez
    • CommentTimeMay 16th 2023

    I used the “center” idea on to show that a compact Hausdorff space can be recovered from the category of complex vector bundles on this space, equipped only with the structure of a ℂ\mathbb{C}-linear category.

    (It’s a lot easier if we treat the category of complex vector bundles as a symmetric monoidal category, but that’s less interesting.)

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2023

    hyperlinked more of the technical terms.

    changed “finite dimensional … vector bundle” to “finite rank …” (or rather to: “finite rank ..”)

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeMay 16th 2023

    Added two remarks on reconstructing a compact Hausdorff space from its category of vector bundles.

    diff, v14, current

    • CommentRowNumber8.
    • CommentAuthorJohn Baez
    • CommentTimeMay 16th 2023

    Added remark:

    Where the center of a category is in general just a commutative monoid (the endomorphism monoid of its identity functor formed in the functor category), for additive categories this commutative monoid carries the further structure of a commutative ring: the endomorphism ring of its identity functor. (In fact this is also true for Ab-enriched categories, which are more general.)

    In fact nothing on this page yet uses any structure beyond Ab-enrichment or enrichment in some category of modules: biproducts play no role. So personally I could change the title of this page to “center of an Ab-enriched category”. But I suspect that historically people studied the additive case and perhaps did more using biproducts.

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2023

    added the definition

    diff, v15, current

    • CommentRowNumber10.
    • CommentAuthorJohn Baez
    • CommentTimeMay 18th 2023

    Improved proposition so it describes the center of the category of modules for any ring RR, not just a commutative one.

    diff, v16, current

1 to 10 of 10

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