The Journal of Symbolic Logic

Research Article

On the category of models of a complete theory

Daniel Lascar 1

Centre National De La Recherche Scientifique, Paris, France

Mcgill University, Montreal, Quebec, Canada

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).

(Received May 09 1979)


1   This work was done while the author was visiting McGill and partially supported by the Natural Sciences and Engineering Research Council of Canada. I would like to thank Michael Makkai for introducing me to categorical logic.