INTRODUCTION The term "categorical logic" has been around for some time now. The expres- sion "categorical model theory" has also been used by several people although it might be harder to find it in the literature. Usually, the latter expression is used to mean something close to "first order categorical logic", the term which served as the title of [M/R]. It is customary to divide categorical logic into a "first order" and a "higher order" part (the core of the latter part of the subject is topos theory). Note that already in [M/R], "first order logic" was meant to incorporate the study of infinitary first order logic, specifically the language L 00 ,w, the infinitary logic with finite quantifiers. As a matter of fact, there are further natural exten- sions to "bigger" infinitary languages, with increased expressive power. In this work, we will be concerned, although often indirectly, with the infinitary logic Loo,oo that allows quantifiers over infinite sets of variables, in addition to the formation rules of Loo,w· The extended interpretation of the term "first order logic" had become established in research in infinitary logic in the sixties and seventies the usage in [M/R] followed that tradition. Under this interpreta- tion, one would use the term "finitary first order logic" to refer to the classical concept. "Model theory" has come to mean the study of ordinary, "set-valued", or en- sembliste models of logical theories. It is by no means axiomatically necessary that studies of first order logic, the latter meant either in a wide or a narrow sense, should focus entirely on set-valued models. In fact, categorical logic shows that the notion of model can be profitably extended in a very general manner to include, e.g., topos-valued models, generalizing Boolean-valued models, etc. Nevertheless, it is a fact that infinitary logic has been pursued with the narrowly construed model-theoretic goals in mind in most of the symbolic-logical litera- ture. The established use of the term "model theory" as the study of set-valued models conveniently focuses the meaning of the term in the way we intend to use it here. Let us then state the meaning of "categorical model theory" as the study of set-valued models of possibly infinitary first order theories by means of the conceptual tools of category theory. Categorical model theory is narrower than categorical logic in general in an- other respect, namely in the fact that it de-emphasizes the role of the theory in favor of the models of the theory. E.g., the Godel completeness theorem, or its categorical versions, are not properly a subject for categorical model theory, although they are a perfectly fitting one for first order categorical logic. The reason is that in the completeness theorem, the internal structure of the under- lying theory has to be taken seriously this internal structure is closely related to syntactical considerations in symbolic logic, and it is not "purely model theo- retical" . These remarks should help to see the difference in orientation between, say, [M/R] and the present work. http://dx.doi.org/10.1090/conm/104
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