More on directed colimits of models. (English) Zbl 0801.18004
Let \(T\) be a finitary first-order theory in a signature \(\Sigma\), \(M(T)\) its category of models (considered with all its homomorphisms). 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.
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.
Reviewer: M.Hébert (Cairo)
MSC:
18A35 | Categories admitting limits (complete categories), functors preserving limits, completions |
18B99 | Special categories |
03C52 | Properties of classes of models |
Citations:
Zbl 0227.02033References:
[1] | J. Ad?mek and J. Rosick?: Reflections in locally presentable categories,Arch. Math. (Brno) 25 (1989), 89-94. |
[2] | J. Ad?mek and J. Rosick?:Locally Presentable and Accessible Categories, Cambridge University Press, 1994. |
[3] | M. A. Dickman:Large Infinitary Languages: Model Theory, North-Holland, 1975. |
[4] | P. C. Eklof and A. M. Mekler:Almost Free Modules, Set-Theoretic Methods, North-Holland, 1990. · Zbl 0718.20027 |
[5] | T. Jech:Set Theory, Academic Press, 1978. · Zbl 0419.03028 |
[6] | M. Makkai and R. Par?: Accessible categories: the foundation of categorical model theory,Contemporary Math. 104, Amer. Math. Soc., 1989. · Zbl 0703.03042 |
[7] | M. Richter: Limites in Kategorien von Relationsystemen,Z. Math. Logik Grundlag. Math. 17 (1971), 75-90. · Zbl 0227.02033 · doi:10.1002/malq.19710170114 |
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