On the category of models of a complete theory

Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object of this paper will be the study of C ( T ). The idea that a better understanding of the category may give us model theoretic information about T is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: “to what extent does this category C ( T ) determine T ?” There is some obvious limitation: for example let T 0 be the theory of infinite sets (in a language containing only =) and T 1 the theory, in the language ( =, U ( ν 0 ), f ( ν 0 )) stating that: (1) U is infinite. (2) f is a bijective map from U onto its complement. It is quite easy to see that C ( T 0 ) is equivalent to C ( T 1 ). But, in this case, T 0 and T 1 can be “interpreted” each in the other. To make this notion of interpretation precise, we shall associate with each theory T a category, loosely denoted by T , defined as follows: (1) The objects are the formulas in the given language. (2) The morphisms from into are the formulas such that (i.e. f defines a map from ϕ into ϕ ; two morphisms defining the same map in all models of T should be identified).