A Functorial Model Theory : Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos

This book is an introduction to a functorial model theory based on infinitary language categories. The author introduces the properties and foundation of these categories before developing a model theory for functors starting with a countable fragment of an infinitary language. He also presents a new technique for generating generic models with categories by inventing infinite language categories and functorial model theory. In addition, the book covers string models, limit models, and functorial models. The book is a preliminary introduction to a functorial model theory based on infinitary language categories. The perspective is different from the preceding authors on the areas in that functorial model theory is based on defining categories on language fragment then carrying on functors to sets and categories to develop models. Infinitary language categories are defined and their preliminary categorical properties are presented. A foundation for infinitary language categories is presented as well. Thus the present mathematics has it own theories and application areas. The book defines a model theory for functors starting with a countable fragment of an infinitary language. The infinite language category defined on L?1,? generic sets on the L?1.K, the respective Kiesler fragment and a functor from L?1,K to Set, called the generic functor are defined. A new technique for generating generic models with categories is defined by inventing infinite language categories and functorial model theory. Specific functor models, the String Models, are defined by infinite chains on fragment models defined on an infinite language category functor. Techniques similar to Robinson's consistency theorem are invented to define limit models. Functorial models are further defined on a generalized diagram by limit chains. The techniques are further developed over the past several years to fill the gap between forcing and toposes with fragment consistent categories Introduction Categorical Preliminaries Categories and Functors Morphisms Functors Categorical Products Natural Transformations Products on Models Preservation of Limits Model Theory and Topoi More on Universal Constructions Chapter Exercises Infinite Language Categories Basics Limits and Infinitary Languages Generic Functors and Language String Models Functorial Morphic Ordered Structure Models Chapter Exercises Functorial Morphic Ordered Structure Models Functorial Fragment M