ABSTRACT
This chapter defines functorial models on ordered structures. Infinite language categories and their preliminary categorical properties are presented. The techniques we have put forth are not the same as a categorical interpretation for logic as in Categorical Logic. What had not been known before “our functorial” can be replaced by “the present Functorial”, is Lω1, ω’s subtle categorical structure and the structure’s significance for computing theory. The categorical structure defined by INFLCS [6] and its Functorial model theory is the foundations for the project. Computable Functors are defined in a December 1996 chapter [11]. There is a basis for a functorial language-structure models. The project allows us specify computation with Lω1, ω as a language for which there are prespecified functorial semantics. There are many chapters on Lω1, ω applications to computing theory. The categorical structure, however, is new and has no obvious connections to the 1980’s applications. The new functorial techniques are applicable to categorical logic where while logic can be carried out at categories, defining models presents us with challenging new mathematics areas. To define solution sets for categories we had developed new techniques, which are in part categorical topology [5]. We start with a preliminary form to the adjoint functor theorem. The solution set conditions
are necessary for presenting initial objects and form the basis for the above preliminary theorem and to what is referred to as the adjoint functor theorem and the basis for many fundamental results in category theory. We had presented techniques [4] for obtaining solution sets by defining a model theory with a countable fragment of an infinitary logic for categories. Further directions for research are presented as a consequence.
