A simple characterization of theories of varieties
type: Item
instance of: scholarly article
title: A simple characterization of theories of varieties
zbMATH Open document ID: 0989.18006
author: R. Smith
published in: Journal of Algebra
publication date: 2002-07-22T00:00:00Z
review text: It is known [\textit{J. Adámek} and \textit{H.-E. Porst}, J. Algebra 208, No. 2, 379-398 (1998; Zbl 0913.18004)] that the theories whose categories of models are quasivarieties are precisely the finitely cocomplete categories with enough regular projectives. It is also known [\textit{A. Carboni, M. C. Peddichio}, and \textit{J. Rosický}, J. Pure Appl. Algebra 161, No. 1-2, 65-90 (2001; Zbl 0982.18006)] that to exact categories regarded as categories of models correspond certain types of theories called ``proexact'' but which are rather technically defined. Since the varieties are the exact quasivarieties, a combination of the above two results readily gives a characterization of the theories of varieties. What the authors show in this paper is that a simpler approach is more fruitful and prove that the theories whose categories of models are the varieties are precisely the finitely cocomplete categories having enough effective projectives, by which they mean objects \(E\) such that \(\hom(E,-)\) preserves coequalizers of reflexive graphs. They also exhibit the 2-category of varieties as a bireflective sub 2-category of the 2-category of locally finitely presentable categories.
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 18C35
Mathematics Subject Classification ID: 18D05
zbMATH DE Number: 1556966
zbMATH Keywords: categories of models
zbMATH Keywords: quasivarieties
zbMATH Keywords: finitely cocomplete categories
zbMATH Keywords: enough regular projectives
zbMATH Keywords: locally finitely presentable categories
reviewed by: Marta Bunge
MaRDI profile type: Publication
full work available at URL: jabr.2000.8447
OpenAlex ID: W2046312765
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DOI: 10.1006/JABR.2000.8447
description: scientific article; zbMATH DE number 1556966