Categories of projective spaces
Starting with an abelian category \({\mathcal A}\), a natural construction [the second author, ``Transfer functors and projective spaces'', Math. Nachr. 118, 147-165 (1984; Zbl 0556.18005)] produces a category \(\mathbb{P} {\mathcal A}\) such that if \({\mathcal A}\) is a category of vector spaces, then \(\mathbb{P} {\mathcal A}\) is the corresponding category of projective spaces. Now the authors introduce an abstract notion of projective category and, generalizing the theory of additive relations, describe an abelianization construction \(\mathbb{A} : \text{(Projective categories)} \to \text{(Abelian categories)}\), so that \(\mathbb{A} \mathbb{P} {\mathcal A}\) is canonically equivalent to \({\mathcal A}\) if \({\mathcal A}\) is abelian, and \(\mathbb{P} \mathbb{A} {\mathcal P}\) is canonically equivalent to \({\mathcal P}\) if \({\mathcal P}\) is projective. This nice ``one-to-one correspondence'' between the abelian and projective categories deeply reflects the relationship between the affine and projective geometries.
scientific article; zbMATH DE number 896574