A Functorial Model Theory

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

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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

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

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Definable Additive Categories: Purity and Model Theory

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

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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

Model Categories
Title Model Categories PDF eBook
Author Mark Hovey
Publisher American Mathematical Soc.
Pages 229
Release 2007
Genre Mathematics
ISBN 0821843613

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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

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

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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

Model Categories and Their Localizations

Model Categories and Their Localizations
Title Model Categories and Their Localizations PDF eBook
Author Philip S. Hirschhorn
Publisher American Mathematical Soc.
Pages 482
Release 2003
Genre Mathematics
ISBN 0821849174

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The aim of this book is to explain modern homotopy theory in a manner accessible to graduate students yet structured so that experts can skip over numerous linear developments to quickly reach the topics of their interest. Homotopy theory arises from choosing a class of maps, called weak equivalences, and then passing to the homotopy category by localizing with respect to the weak equivalences, i.e., by creating a new category in which the weak equivalences are isomorphisms. Quillen defined a model category to be a category together with a class of weak equivalences and additional structure useful for describing the homotopy category in terms of the original category. This allows you to make constructions analogous to those used to study the homotopy theory of topological spaces. A model category has a class of maps called weak equivalences plus two other classes of maps, called cofibrations and fibrations. Quillen's axioms ensure that the homotopy category exists and that the cofibrations and fibrations have extension and lifting properties similar to those of cofibration and fibration maps of topological spaces. During the past several decades the language of model categories has become standard in many areas of algebraic topology, and it is increasingly being used in other fields where homotopy theoretic ideas are becoming important, including modern algebraic $K$-theory and algebraic geometry. All these subjects and more are discussed in the book, beginning with the basic definitions and giving complete arguments in order to make the motivations and proofs accessible to the novice. The book is intended for graduate students and research mathematicians working in homotopy theory and related areas.

Category Theory in Context

Category Theory in Context
Title Category Theory in Context PDF eBook
Author Emily Riehl
Publisher Courier Dover Publications
Pages 273
Release 2017-03-09
Genre Mathematics
ISBN 0486820807

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Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.