1st Edition

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

ISBN 9781774633106
Published March 31, 2021 by Apple Academic Press
302 Pages 25 B/W Illustrations

USD $54.95

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Book Description

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.

Table of Contents


Categorical Preliminaries
Categories and Functors
Categorical Products
Natural Transformations
Products on Models
Preservation of Limits
Model Theory and Topoi
More on Universal Constructions
Chapter Exercises

Infinite Language Categories
Limits and Infinitary Languages
Generic Functors and Language String Models
Functorial Morphic Ordered Structure Models
Chapter Exercises
Functorial Morphic Ordered Structure Models

Functorial Fragment Model Theory
Generic Functors and Language String Models
Functorial Models As ¿-Chains
Models Glimpses From Functors
Structure Products
Higher Stratified Consistency and Completeness
Fragment Positive Omitting Type Algebras
Omitting Types and Realizability
Positive Categories and Consistency Models
More on Fragment Consistency
Chapter Exercises

Algebraic Theories, Categories, and Models
Ultraproducts on Algebras
Ultraproducts and Ultrafilters
Ultraproduct Applications to Horn Categories
Algebraic Theories and Topos Models
Free Theories and Factor Theories
T-Algebras and Adjunctions
Theory Morphisms, Products and Co-products
Algebras and the Category of Algebraic Theories
Initial Algebraic Theories and Computable Trees
Chapter Exercises

Generic Functorial Models and Topos
Elementary Topoi
Generic Functorial Models
Generic Functors
Initial D<A,G> Models
Positive Forcing Models
Functors Computing Hasse Diagram Models
Fragment Consistent Models
Homotopy theory of topos
Filtered colimits and comma categories
More on Yoneda Lemma
Chapter Exercises

Models, Sheaves, and Topos
Duality, Fragment Models, and Topology
Lifts on Topos Models on Cardinalities
Chapter Exercises

Functors on Fields
Basic Models
Prime Models
Omitting Types on Fields
Filters and Fields
Filters and Products
Chapter Exercises

Filters and Ultraproducts on Projective Sets
General Definitions
Generic Functors and Language String Models
Functorial Fragment Consistency
Structure Products
Completing Theories and Fragments
Prime Models and Model Completion
Uniform and countably incomplete ultrafilters
Functorial Projetive Set Models and Saturation
Ultraproducts and Ultrafliters
Chapter Exercises

A Glimpse on m Algebraic Set Theory
Ultraproducts and Ultrafilters on Sets
Ultrafilters over N
Saturation and Preservations
Functorial Models and Descriptive Sets
Filters, Fragment Constructible Models, and Sets

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Dr. Cyrus F. Nourani is a consultant in computing R&D and a research professor at Simon Fraser University. He has many years of experience in the design and implementation of computing systems and has authored/coauthored several books and over 350 publications in mathematics and computer science. He has also held faculty positions at numerous institutions, including the University of Michigan, University of Pennsylvania, University of Auckland, UCLA, and MIT. His research interests include computer science, artificial intelligence, mathematics, virtual haptic computation, information technology, and management.