Unexpected properties of locally presentable categories
type: Item
instance of: scholarly article
title: Unexpected properties of locally presentable categories
zbMATH Open document ID: 0701.18003
DOI: 10.1007/BF01182450
author: Jiří Rosický
author: Jiří Adámek
author: Věra Trnková
published in: Algebra Universalis
publication date: 1990-01-01T00:00:00Z
review text: Let \({\mathcal O}rd\) be the usual category of all ordinals and \({\mathcal G}ra\) the category of all graphs \([=binary\) relations]. The statement that \({\mathcal O}rd\) cannot be fully embedded into \({\mathcal G}ra\) is called Vopěnka's principle. Locally presentable categories are precisely those complete categories which have a small dense subcategory (all subcategories are meant to be full in this paper). The main result is the equivalence of Vopěnka's principle to various properties of locally presentable categories, for example to each of the following three claims: every subcategory of a locally presentable category (i) has a small dense subcategory; (ii) if it is closed under limits, then it is locally presentable; (iii) if it is closed under colimits, then it is coreflective.
Mathematics Subject Classification ID: 18B15
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E65
zbMATH DE Number: 4148378
zbMATH Keywords: Vopěnka's principle
zbMATH Keywords: locally presentable categories
zbMATH Keywords: small dense subcategory
MaRDI profile type: Publication
cites work: Q4723930
cites work: Adequate subcategories
cites work: Q4693151
cites work: Accessible Categories: The Foundations of Categorical Model Theory
cites work: A note on reflection maps
cites work: Q3672155
cites work: Q3208855
description: scientific article; zbMATH DE number 4148378