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On the category of models of a complete theory

  • Daniel Lascar (a1) (a2)

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 T0 be the theory of infinite sets (in a language containing only =) and T1 the theory, in the language ( =, U(ν0),f(ν0)) 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(T0) is equivalent to C(T1). But, in this case, T0 and T1 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).

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[LP]Lascar D. and Poizat B., An introduction to forking, this Journal, vol. 44 (1979), pp. 330350.
[M1]Makkai M., Full continuous embeddings of toposes (to appear).
[M2]Makkai M., The topos of types (to appear).
[MR]Makkai M. and Reyes G., First order categorical logic, Lecture Notes in Mathematics, vol. 611, Springer-Verlag, Berlin and New York.
[S]Shelah S., Classification theory and the number of non-isomorphic models, Studies in logic, North-Holland, Amsterdam, 1978.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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