## Introduction to Higher-Order Categorical Logic

**Author**: J. Lambek,P. J. Scott

**Publisher:**Cambridge University Press

**ISBN:**9780521356534

**Category:**Mathematics

**Page:**304

**View:**2695

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Part I indicates that typed-calculi are a formulation of higher-order logic, and cartesian closed categories are essentially the same. Part II demonstrates that another formulation of higher-order logic is closely related to topos theory.

## Categorical Logic and Type Theory

**Author**: Bart Jacobs

**Publisher:**Gulf Professional Publishing

**ISBN:**9780444508539

**Category:**Mathematics

**Page:**760

**View:**4973

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This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

## Categories for Types

**Author**: Roy L. Crole

**Publisher:**Cambridge University Press

**ISBN:**9780521457019

**Category:**Computers

**Page:**335

**View:**4683

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This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.

## Higher Topos Theory (AM-170)

**Author**: Jacob Lurie

**Publisher:**Princeton University Press

**ISBN:**1400830559

**Category:**Mathematics

**Page:**944

**View:**8198

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Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

## Theorem Proving in Higher Order Logics

*22nd International Conference, TPHOLs 2009, Munich, Germany, August 17-20, 2009, Proceedings*

**Author**: Stefan Berghofer,Tobias Nipkow,Christian Urban,Makarius Wenzel

**Publisher:**Springer

**ISBN:**3642033598

**Category:**Computers

**Page:**517

**View:**1490

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This book constitutes the refereed proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics, TPHOLs 200, held in Munich, Germany, in August 2009. The 26 revised full papers presented together with 1 proof pearl, 4 tool presentations, and 3 invited papers were carefully reviewed and selected from 55 submissions. The papers cover all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification such as formal semantics of specification, modeling, and programming languages, specification and verification of hardware and software, formalization of mathematical theories, advances in theorem prover technology, as well as industrial application of theorem provers.

## Generic Programming

*Advanced Lectures*

**Author**: Roland Backhouse,Jeremy Gibbons

**Publisher:**Springer Science & Business Media

**ISBN:**3540201947

**Category:**Computers

**Page:**221

**View:**6604

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Generic programming attempts to make programming more efficient by making it more general. This book is devoted to a novel form of genericity in programs, based on parameterizing programs by the structure of the data they manipulate. The book presents the following four revised and extended chapters first given as lectures at the Generic Programming Summer School held at the University of Oxford, UK in August 2002: - Generic Haskell: Practice and Theory - Generic Haskell: Applications - Generic Properties of Datatypes - Basic Category Theory for Models of Syntax

## Mathematical Logic and Theoretical Computer Science

**Author**: Kueker

**Publisher:**CRC Press

**ISBN:**9780824777463

**Category:**Mathematics

**Page:**408

**View:**4493

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## Logic and Computation

*Interactive Proof with Cambridge LCF*

**Author**: Lawrence C. Paulson

**Publisher:**Cambridge University Press

**ISBN:**9780521395601

**Category:**Computers

**Page:**320

**View:**9256

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Logic and Computation is concerned with techniques for formal theorem-proving, with particular reference to Cambridge LCF (Logic for Computable Functions). Cambridge LCF is a computer program for reasoning about computation. It combines methods of mathematical logic with domain theory, the basis of the denotational approach to specifying the meaning of statements in a programming language. This book consists of two parts. Part I outlines the mathematical preliminaries: elementary logic and domain theory. They are explained at an intuitive level, giving references to more advanced reading. Part II provides enough detail to serve as a reference manual for Cambridge LCF. It will also be a useful guide for implementors of other programs based on the LCF approach.

## Automated Reasoning

*Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17-20, 2006, Proceedings*

**Author**: Ulrich Furbach,Natarajan Shankar

**Publisher:**Springer Science & Business Media

**ISBN:**3540371877

**Category:**Computers

**Page:**688

**View:**6484

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Here are the proceedings of the Third International Joint Conference on Automated Reasoning, IJCAR 2006, held in Seattle, Washington, USA, August 2006. The book presents 41 revised full research papers and 8 revised system descriptions, with 3 invited papers and a summary of a systems competition. The papers are organized in topical sections on proofs, search, higher-order logic, proof theory, proof checking, combination, decision procedures, CASC-J3, rewriting, and description logic.

## Basic Category Theory

**Author**: Tom Leinster

**Publisher:**Cambridge University Press

**ISBN:**1107044243

**Category:**Mathematics

**Page:**190

**View:**8730

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A short introduction ideal for students learning category theory for the first time.

## Galois Theories

**Author**: Francis Borceux,George Janelidze

**Publisher:**Cambridge University Press

**ISBN:**9780521803090

**Category:**Mathematics

**Page:**341

**View:**7798

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Develops Galois theory in a more general context, emphasizing category theory.

## An Introduction to Invariants and Moduli

**Author**: Shigeru Mukai,W. M. Oxbury

**Publisher:**Cambridge University Press

**ISBN:**9780521809061

**Category:**Mathematics

**Page:**503

**View:**9629

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Incorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, it's influence is not confined there; for example the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. An accurate account of Mukai's influential Japanese texts, this tranlation will be a valuable resource for researchers and graduate students working in a range of areas.

## Conceptual Mathematics

*A First Introduction to Categories*

**Author**: F. William Lawvere,Stephen H. Schanuel

**Publisher:**Cambridge University Press

**ISBN:**0521894859

**Category:**Mathematics

**Page:**390

**View:**9562

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In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.

## Notes on Logic and Set Theory

**Author**: P. T. Johnstone

**Publisher:**Cambridge University Press

**ISBN:**9780521336925

**Category:**Mathematics

**Page:**110

**View:**914

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A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics.

## Practical Foundations of Mathematics

**Author**: Paul Taylor

**Publisher:**Cambridge University Press

**ISBN:**9780521631075

**Category:**Mathematics

**Page:**572

**View:**5793

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Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.

## Lecture notes in pure and applied mathematics

**Author**: N.A

**Publisher:**N.A

**ISBN:**N.A

**Category:**Mathematics

**Page:**N.A

**View:**8994

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## A Primer of Nonlinear Analysis

**Author**: Antonio Ambrosetti,Giovanni Prodi

**Publisher:**Cambridge University Press

**ISBN:**9780521485739

**Category:**Mathematics

**Page:**171

**View:**6400

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This is an elementary and self-contained introduction to nonlinear functional analysis and its applications, especially in bifurcation theory.

## Categorical Logic

**Author**: Andrew M. Pitts

**Publisher:**N.A

**ISBN:**N.A

**Category:**Logic, Symbolic and mathematical

**Page:**94

**View:**2826

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Abstract: "This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists."

## Relative category theory and geometric morphisms

*a logical approach*

**Author**: Jonathan Chapman,Frederick Rowbottom

**Publisher:**Oxford University Press, USA

**ISBN:**N.A

**Category:**Mathematics

**Page:**263

**View:**9700

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Topos theory provides an important setting and language for much of mathematical logic and set theory. It is well known that a typed language can be given for a topos to be regarded as a category of sets. This enables a fruitful interplay between category theory and set theory. However, one stumbling block to a logical approach to topos theory has been the treatment of geometric morphisms. This book presents a convenient and natural solution to this problem by developing the notion of a frame relative to an elementary topos. The authors show how this technique enables a logical approach to be taken to topics such as category theory relative to a topos and the relative Giraud theorem. The work is self-contained except that the authors presuppose a familiarity with basic category theory and topos theory. Logicians, set and category theorists, and computer scientist working in the field will find this work essential reading.