Syntactic characterizations of closure under pullbacks and of locally polypresentable categories (Q676312)

In the paper a characterization of a theory \(T\), which is invariant under pullbacks, is given by several equivalent conditions. One of them is that \(T\) has polypresentations. A theory \(T\) is said to be invariant under pullbacks, iff the category of models of \(T\) is closed under the construction of pullbacks. The theory of algebraically closed fields is invariant under pullbacks but not under connected limits. As a second result it is shown that any category is locally \(\omega\)-polypresentable if and only if it is equivalent to the category of models of a pullback theory. Finally it is proved that the category of models of a finitary theory which is invariant under equalizers is not in general finitely accessible.