Toposes, Triples And Theories

This book is an introduction to toposes, triples and theories and the connections between them.

**Tag(s):**
Category Theory

**Publication date**: 30 Nov -0001

**ISBN-10**:
n/a

**ISBN-13**:
n/a

**Paperback**:
n/a

**Views**: 17,789

**Type**: N/A

**Publisher**:
n/a

**License**:
n/a

**Post time**: 09 Dec 2009 05:21:52

Toposes, Triples And Theories

This book is an introduction to toposes, triples and theories and the connections between them.

License Reminder:

Excerpts from the Preface:

Michael Barr wrote:Copyright 2000 by Michael Barr and Charles Frederick Wells. This version may be downloaded and printed in unmodified form for private use only.

Excerpts from the Preface:

Michael Barr wrote:As its title suggests, this book is an introduction to three ideas and the connections between them. Before describing the content of the book in detail, we describe each concept briely. More extensive introductory descriptions of each concept are in the introductions and notes to Chapters 2, 3 and 4.

A topos is a special kind of category defined by axioms saying roughly that certain constructions one can make with sets can be done in the category. In that sense, a topos is a generalized set theory. However, it originated with Grothendieck and Giraud as an abstraction of the properties of the category of sheaves of sets on a topological space. Later, Lawvere and Tierney introduced a more general idea which they called "elementary topos" (because their axioms were first order and involved no quantification over sets), and they and other mathematicians developed the idea that a theory in the sense of mathematical logic can be regarded as a topos, perhaps after a process of completion.

The concept of triple originated (under the name "standard constructions") in Godement's book on sheaf theory for the purpose of computing sheaf cohomology. Then Peter Huber discovered that triples capture much of the information of adjoint pairs. Later Linton discovered that triples gave an equivalent approach to Lawvere's theory of equational theories (or rather the infinite generalizations of that theory). Finally, triples have turned out to be a very important tool for deriving various properties of toposes.

Theories, which could be called categorical theories, have been around in one incarnation or another at least since Lawvere's Ph.D. thesis. Lawvere's original insight was that a mathematical theory--corresponding roughly to the definition of a class of mathematical objects--could be usefully regarded as a category with structure of a certain kind, and a model of that theory--one of those objects--as a set-valued functor from that category which preserves the structure. The structures involved are more or less elaborate, depending on the kind of objects involved. The most elaborate of these use categories which have all the structure of a topos.

Tweet

About The Author(s)

Peter Redpath Emeritus Professor of Pure Mathematics, Department of Mathematics and Statistics, McGill University.

No information is available for this author.

Book Categories

Computer Science
Introduction to Computer Science
Introduction to Computer Programming
Algorithms and Data Structures
Artificial Intelligence
Computer Vision
Machine Learning
Neural Networks
Game Development and Multimedia
Data Communication and Networks
Coding Theory
Computer Security
Information Security
Cryptography
Information Theory
Computer Organization and Architecture
Operating Systems
Image Processing
Parallel Computing
Concurrent Programming
Relational Database
Document-oriented Database
Data Mining
Big Data
Data Science
Digital Libraries
Compiler Design and Construction
Functional Programming
Logic Programming
Object Oriented Programming
Formal Methods
Software Engineering
Agile Software Development
Information Systems
Geographic Information System (GIS)

Mathematics
Mathematics
Algebra
Abstract Algebra
Linear Algebra
Number Theory
Numerical Methods
Precalculus
Calculus
Differential Equations
Category Theory
Proofs
Discrete Mathematics
Theory of Computation
Graph Theory
Real Analysis
Complex Analysis
Probability
Statistics
Game Theory
Queueing Theory
Operations Research
Computer Aided Mathematics

Supporting Fields
Web Design and Development
Mobile App Design and Development
System Administration
Cloud Computing
Electric Circuits
Embedded System
Signal Processing
Integration and Automation
Network Science
Project Management

Operating System
Programming/Scripting
Ada
Assembly
C / C++
Common Lisp
Forth
Java
JavaScript
Lua
Microsoft .NET
Rexx
Perl
PHP
Python
R
Rebol
Ruby
Scheme
Tcl/Tk

Miscellaneous
Sponsors