Embeddings of Exactly Definable and Finitely Accessible Additive Categories into Freyd Categories

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Let ℰ be an additive category and 𝒞 a full subcategory with split idempotents, and closed under isomorphic images and finite direct sums. We give conditions on ℰ and 𝒞 implying that ℰ embeds into an abelian category, so that the objects of 𝒞 turn into injective objects. This construction generalizes the embedding of exactly definable categories into locally coherent categories, while the dual construction generalizes the embedding of finitely accessible categories into Grothendieck categories with a family of finitely generated projective generators. As applications, we characterize exactly definable categories through intrinsic properties and study those locally coherent categories whose fp-injective objects form a Grothendieck category.