We give a syntactic characterization of (finitary) theories whose categories of models are closed under the formation of connected limits (respectively the formation of pullbacks and substructures) in the category of all structures. They are also those theories whose consistent extensions by new atomic facts admit in each component an initial structure (respectively an initial term structure), and also thoseT for whichM(T) is locally finitely multi-presentable in a canonical way. We also show that these two properties of theories are nonuniform.

We give syntactic characterizations of1. the theories whose categories of models are closed under the formation of pullbacks, and of2. the locally ω-polypresentable categories.A somewhat typical example is the category of algebraically closed fields. Case is proved by classical model-theoretic methods; it solves a problem raised by H. Volger . The solution of case 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 was raised in (...) the context of Domain Theory by F. Lamarche. (shrink)