Categories of projective spaces
type: Item
instance of: scholarly article
title: Categories of projective spaces
zbMATH Open document ID: 0863.18006
DOI: 10.1016/0022-4049(95)00111-5
author: A. Carboni
author: Marco Grandis
published in: Journal of Pure and Applied Algebra
publication date: 1997-06-10T00:00:00Z
review text: 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.
reviewed by: George Janelidze
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 51N15
Mathematics Subject Classification ID: 18F99
Mathematics Subject Classification ID: 51N10
zbMATH DE Number: 896574
zbMATH Keywords: exact category
zbMATH Keywords: affine geometries
zbMATH Keywords: abelian category
zbMATH Keywords: category of projective spaces
zbMATH Keywords: projective category
zbMATH Keywords: projective geometries
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00111-5
OpenAlex ID: W2072055337
cites work: Categories of affine spaces
cites work: Matrices, relations, and group representations
cites work: Cartesian bicategories. I
cites work: Q5599870
cites work: Transfer Functors and Projective Spaces
cites work: Q3698980
cites work: Q5682456
cites work: Q5508993
cites work: Korrespondenzen in abelschen Kategorien
description: scientific article; zbMATH DE number 896574