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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:
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.
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