On preaccessible categories
type: Item
instance of: scholarly article
title: On preaccessible categories
zbMATH Open document ID: 0852.18003
DOI: 10.1016/0022-4049(94)00152-9
author: Jiří Adámek
author: Jiří Rosický
published in: Journal of Pure and Applied Algebra
publication date: 1996-05-19T00:00:00Z
review text: For a regular cardinal \(\lambda\), a category is \(\lambda\)-preaccessible if it has a small set \(G\) of \(\lambda\)-presentable objects such that every object is a \(\lambda\)-directed colimit of objects in \(G\); hence this notion drops the required existence of all \(\lambda\)-directed colimits from the familiar notion of \(\lambda\)-accessibility. An easy example of a preaccessible but not accessible category is provided by any small category in which idempotents do not split. A preaccessible category has a dense generating set, and the converse proposition holds true if (and only if) the set-theoretic Vopěnka principle holds true. This principle makes every preaccessible category with equalizers accessible and therefore axiomatizable [in the sense of \textit{M. Makkai} and \textit{R. Paré}, ``Accessible categories: The foundations of categorical model theory'', Contemp. Math. 104 (1989; Zbl 0703.03042)], but it is not known whether every axiomatizable category is preaccessible.
reviewed by: Walter Tholen
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 18C99
zbMATH DE Number: 858618
zbMATH Keywords: regular cardinal
zbMATH Keywords: accessible category
zbMATH Keywords: preaccessible category
zbMATH Keywords: Vopěnka principle
zbMATH Keywords: axiomatizable category
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00152-9
OpenAlex ID: W2087944232
cites work: Q4293501
cites work: Q3852172
cites work: Q3672155
cites work: On the role of supercompact and extendible cardinals in logic
cites work: Accessible Categories: The Foundations of Categorical Model Theory
cites work: More on directed colimits of models
cites work: Unexpected properties of locally presentable categories
description: scientific article; zbMATH DE number 858618