More on directed colimits of models
type: Item
instance of: scholarly article
title: More on directed colimits of models
zbMATH Open document ID: 0801.18004
DOI: 10.1007/BF00878503
published in: Applied Categorical Structures
publication date: 1994-08-10T00:00:00Z
review text: Let \(T\) be a finitary first-order theory in a signature \(\Sigma\), \(M(T)\) its category of models (considered with all its homomorphisms). \textit{M. Richter} [Z. Math. Logik Grundl. Math. 17, 75-90 (1971; Zbl 0227.02033)] showed that if \(M(T)\) has directed colimits, then they are the ``natural'' ones (i.e., the ones in the category \(\text{Str} (\Sigma)\) of all \(\Sigma\)-structures). This paper gives a new elegant proof of this result which can be generalized to infinitary cases under the set- theoretical assumption of the existence of sufficiently large compact cardinals. The context is in fact more general: one can replace \(M(T)\) by the category \(\text{Red}_{\Sigma', \Sigma} T\) of \(\Sigma\)-reducts (with \(\Sigma\)-homomorphisms) of the models of a \(L_ \lambda (\Sigma')\)- theory \(T\) (where \(\Sigma\subseteq \Sigma')\), and then the inclusion functor of \(\text{Red}_{\Sigma', \Sigma} T\) into \(\text{Str}(\Sigma)\) is shown to preserve \(\kappa\)-directed colimits, where \(\kappa\) is a compact cardinal \(\geq\lambda\). The paper also characterizes such categories of reducts as precisely the full images of accessible functors. As a consequence, the equivalence, for an accessible category \(K\), between the three properties: locally presentable, complete and cocomplete, also holds if \(K\) is the full image of an accessible functor, provided there exists a proper class of compact cardinals. Whether this result and the first one hold without set- theoretical assumptions is unknown.
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 18B99
Mathematics Subject Classification ID: 03C52
zbMATH DE Number: 617352
zbMATH Keywords: directed colimits
zbMATH Keywords: elementary classes of models
zbMATH Keywords: accessible categories
zbMATH Keywords: compact cardinals
author: Jiří Rosický
reviewed by: Michel Hébert
MaRDI profile type: Publication
cites work: Q3995749
cites work: Q3852172
cites work: Accessible Categories: The Foundations of Categorical Model Theory
cites work: Limites in Kategorien von Relationalsystemen
full work available at URL: https://doi.org/10.1007/bf00878503
OpenAlex ID: W2147810853
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description: scientific article; zbMATH DE number 617352