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 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 nforum nlab noncommutative noncommutative-geometry number-theory 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 sheaves 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: center of an additive category

Bottom of Page

1 to 10 of 10

  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

  1.