Syntactic characterizations of closure under pullbacks and of locally polypresentable categories

Abstract

We give syntactic characterizations of 1. (1) the (finitary) theories whose categories of models are closed under the formation of pullbacks, and of 2. (2) (its categorical counterpart) the locally ω-polypresentable categories. A somewhat typical example is the category of algebraically closed fields. Case (1) is proved by classical model-theoretic methods; it solves a problem raised by H. Volger (with motivations from the theory of abstract data types). The solution of case (2) is in the spirit of the ones for the locally ω-presentable and ω-multipresentable cases found by M. Coste and P.T. Johnstone respectively. The problem (2) was raised in the context of Domain Theory by F. Lamarche.