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

]]>added the definition

]]>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.

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

]]>hyperlinked more of the technical terms.

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

]]>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 $\mathbb{C}$-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.)

]]>Added remark: why Morita equivalent commutative rings are isomorphic.

]]>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.)

]]>added more hyperlinking and formatting (here)

]]>Showed that for any commutative ring $R$, the center of the category of $R$-modules is $R$.

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