eBook ISBN:  9781470400804 
Product Code:  MEMO/105/503.E 
List Price:  $38.00 
MAA Member Price:  $34.20 
AMS Member Price:  $22.80 
eBook ISBN:  9781470400804 
Product Code:  MEMO/105/503.E 
List Price:  $38.00 
MAA Member Price:  $34.20 
AMS Member Price:  $22.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 105; 1993; 106 ppMSC: Primary 03; 18
Using the theory of categories as a framework, this book develops a duality theory for theories in first order logic in which the dual of a theory is the category of its models with suitable additional structure. This duality theory resembles and generalizes M. H. Stone's famous duality theory for Boolean algebras. As an application, Makkai derives a result akin to the wellknown definability theorem of E. W. Beth. This new definability theorem is related to theorems of descent in category theory and algebra and can also be stated as a result in pure logic without reference to category theory. Containing novel techniques as well as applications of classical methods, this carefully written book shows attention to both organization and detail and will appeal to mathematicians and philosophers interested in category theory.
ReadershipMathematicians and philosophers interested in category theory and mathematical logic.

Table of Contents

Chapters

1. Beth’s theorem for propositional logic

2. Factorizations in 2categories

3. Definable functors

4. Basic notions for duality

5. The Stonetype adjunction for Boolean pretoposes and ultragroupoids

6. The syntax of special ultramorphisms

7. The semantics of special ultramorphisms

8. The duality theorem

9. Preparing a functor specification

10. Lifting Zawadowski’s argument to ultra*morphisms

11. The operations in $\mathcal {BP}^*$ and $\mathtt {UG}$

12. Conclusion


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Using the theory of categories as a framework, this book develops a duality theory for theories in first order logic in which the dual of a theory is the category of its models with suitable additional structure. This duality theory resembles and generalizes M. H. Stone's famous duality theory for Boolean algebras. As an application, Makkai derives a result akin to the wellknown definability theorem of E. W. Beth. This new definability theorem is related to theorems of descent in category theory and algebra and can also be stated as a result in pure logic without reference to category theory. Containing novel techniques as well as applications of classical methods, this carefully written book shows attention to both organization and detail and will appeal to mathematicians and philosophers interested in category theory.
Mathematicians and philosophers interested in category theory and mathematical logic.

Chapters

1. Beth’s theorem for propositional logic

2. Factorizations in 2categories

3. Definable functors

4. Basic notions for duality

5. The Stonetype adjunction for Boolean pretoposes and ultragroupoids

6. The syntax of special ultramorphisms

7. The semantics of special ultramorphisms

8. The duality theorem

9. Preparing a functor specification

10. Lifting Zawadowski’s argument to ultra*morphisms

11. The operations in $\mathcal {BP}^*$ and $\mathtt {UG}$

12. Conclusion