A Functorial Model Theory
Title | A Functorial Model Theory PDF eBook |
Author | Cyrus F. Nourani |
Publisher | CRC Press |
Pages | 296 |
Release | 2016-04-19 |
Genre | Mathematics |
ISBN | 1482231506 |
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.
Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods
Title | Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods PDF eBook |
Author | Alexander Martsinkovsky |
Publisher | Springer Nature |
Pages | 256 |
Release | |
Genre | |
ISBN | 3031530632 |
Definable Additive Categories: Purity and Model Theory
Title | Definable Additive Categories: Purity and Model Theory PDF eBook |
Author | Mike Prest |
Publisher | American Mathematical Soc. |
Pages | 122 |
Release | 2011-02-07 |
Genre | Mathematics |
ISBN | 0821847678 |
Most of the model theory of modules works, with only minor modifications, in much more general additive contexts (such as functor categories, categories of comodules, categories of sheaves). Furthermore, even within a given category of modules, many subcategories form a ``self-sufficient'' context in which the model theory may be developed without reference to the larger category of modules. The notion of a definable additive category covers all these contexts. The (imaginaries) language which one uses for model theory in a definable additive category can be obtained from the category (of structures and homomorphisms) itself, namely, as the category of those functors to the category of abelian groups which commute with products and direct limits. Dually, the objects of the definable category--the modules (or functors, or comodules, or sheaves)--to which that model theory applies may be recovered as the exact functors from the, small abelian, category (the category of pp-imaginaries) which underlies that language.
Model Categories
Title | Model Categories PDF eBook |
Author | Mark Hovey |
Publisher | American Mathematical Soc. |
Pages | 229 |
Release | 2007 |
Genre | Mathematics |
ISBN | 0821843613 |
Model categories are used as a tool for inverting certain maps in a category in a controllable manner. They are useful in diverse areas of mathematics. This book offers a comprehensive study of the relationship between a model category and its homotopy category. It develops the theory of model categories, giving a development of the main examples.
Algebraic Computability and Enumeration Models
Title | Algebraic Computability and Enumeration Models PDF eBook |
Author | Cyrus F. Nourani |
Publisher | CRC Press |
Pages | 304 |
Release | 2016-02-24 |
Genre | Mathematics |
ISBN | 1771882484 |
This book, Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic viewpoint. The reader is first introduced to categories and functorial models, with Kleene algebra examples
Accessible Categories: The Foundations of Categorical Model Theory
Title | Accessible Categories: The Foundations of Categorical Model Theory PDF eBook |
Author | Mihály Makkai |
Publisher | American Mathematical Soc. |
Pages | 186 |
Release | 1989 |
Genre | Mathematics |
ISBN | 082185111X |
Intended for category theorists and logicians familiar with basic category theory, this book focuses on categorical model theory, which is concerned with the categories of models of infinitary first order theories, called accessible categories. The starting point is a characterization of accessible categories in terms of concepts familiar from Gabriel-Ulmer's theory of locally presentable categories. Most of the work centers on various constructions (such as weighted bilimits and lax colimits), which, when performed on accessible categories, yield new accessible categories. These constructions are necessarily 2-categorical in nature; the authors cover some aspects of 2-category theory, in addition to some basic model theory, and some set theory. One of the main tools used in this study is the theory of mixed sketches, which the authors specialize to give concrete results about model theory. Many examples illustrate the extent of applicability of these concepts. In particular, some applications to topos theory are given. Perhaps the book's most significant contribution is the way it sets model theory in categorical terms, opening the door for further work along these lines. Requiring a basic background in category theory, this book will provide readers with an understanding of model theory in categorical terms, familiarity with 2-categorical methods, and a useful tool for studying toposes and other categories.
Categorical Homotopy Theory
Title | Categorical Homotopy Theory PDF eBook |
Author | Emily Riehl |
Publisher | Cambridge University Press |
Pages | 371 |
Release | 2014-05-26 |
Genre | Mathematics |
ISBN | 1139952633 |
This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.