Localizations of algebraic categories. II
type: Item
instance of: scholarly article
title: Localizations of algebraic categories. II
zbMATH Open document ID: 0939.18006
DOI: 10.1016/S0022-4049(97)00122-9
author: E. M. Vitale
published in: Journal of Pure and Applied Algebra
publication date: 2000-07-12T00:00:00Z
review text: The author had previously characterized the localizations of monadic categories over \({\mathcal S}{\mathcal E}{\mathcal T}\) as the Barr-exact categories with a regular generator and all of its copowers [see part I, \textit{E. M. Vitale}, J. Pure Appl. Algebra 108, No. 3, 315-320 (1996; Zbl 0854.18007)]. By adding to these conditions the requirement that directed colimits commute with finite limits, he has now obtained a characterization of localizations of algebraic categories in the sense of Lawvere. This result also leads to a new proof of the Popescu-Gabriel characterization of Grothendieck categories: the abelian categories with a generator and its copowers in which directed colimits exist and commute with finite limits are precisely the localizations of module categories.
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1308284
zbMATH Keywords: exact categories
zbMATH Keywords: monadic categories
zbMATH Keywords: localizations
zbMATH Keywords: algebraic categories
zbMATH Keywords: Grothendieck categories
reviewed by: Walter Tholen
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00122-9
OpenAlex ID: W2009828041
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description: scientific article; zbMATH DE number 1308284