Toposes, Triples and Theories
By: by Michael Barr, Charles Wells
Contributors: Wells, Charles, author
Series: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics -- 278; Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics -- 278.
Publisher: Springer New York; Springer
Published: 1985
Place of publication: New York, NY
Language: eng
Number of pages: 347
ISBN-13: 9781489900210; 9781489900234
Subjects: K-theory; Mathematics; Symbolic and mathematical Logic; Mathematical Logic and Foundations
Description: 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 briefly. 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 of the category of sheaves of sets on a topological space. Later, properties Lawvere and Tierney introduced a more general id~a which they called "elementary topos" (because their axioms did not quantify 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 construc­ in Godement's book on sheaf theory for the purpose of computing tions") 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 Lawverc'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.
Table of contents: title: 1. Categories; title: 2. Toposes; title: 3. Triples; title: 4. Theories; title: 5. Properties of Toposes; title: 6. Permanence Properties of Toposes; title: 7. Representation Theorems; title: 8. Cocone Theories; title: 9. More on Triples; title: Index to Exercises.