More on directed colimits of models (Q1330914)

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.