Separable K-Linear Categories

We define and investigate separable K-linear categories. We show that such a category C is locally finite and that every left C-module is projective. We apply our main results to characterize separable linear categories that are spanned by groupoids or delta categories.

💡 Research Summary

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The paper introduces and studies the notion of a separable K‑linear category C. A K‑linear category is called separable if its composition (multiplication) functor μ : C ⊗ₖ C → C admits a splitting as a C‑C‑bimodule map. This definition mirrors the classical one for separable K‑algebras, where the multiplication map A ⊗ₖ A → A splits as an A‑A‑bimodule homomorphism.

The authors first establish two fundamental structural consequences of separability. Theorem 2.3 shows that any separable K‑linear category is locally finite: for every pair of objects x, y the hom‑space Hom_C(x, y) is a finite‑dimensional K‑vector space. The proof uses the existence of a bimodule section s : C → C ⊗ₖ C to write C as a direct sum of finitely many sub‑bimodules, forcing each hom‑space to be finite‑dimensional.

The second main result (Theorem 3.1) proves that every left C‑module is projective. Because the splitting s makes C a projective C‑C‑bimodule, any left module M can be expressed as a retract of a free module C ⊗ₖ V for some K‑vector space V. Consequently, the category of left C‑modules is semisimple: there are no non‑trivial extensions and every module decomposes as a direct sum of simple modules. This property is a categorical analogue of the classical fact that a separable algebra is semisimple.

Having set up the general theory, the paper turns to two concrete families of K‑linear categories.

1. Groupoid linearizations.
   Let G be a (small) groupoid and K