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We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer’s prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked to Serre prime spectra of \(\mathbb {Z}_+\)-rings. As an application, we construct a categorification of the quantized coordinate rings of open Richardson varieties for symmetric Kac–Moody groups, by constructing Serre completely prime ideals of monoidal categories of modules of the KLR algebras, and by taking Serre quotients with respect to them.

Prime Spectra of Abelian 2-Categories and Categorifications of Richardson Varieties