*Steve Awodey*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.001.0001
- Subject:
- Mathematics, Algebra

This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible ...
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This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.Less

This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.

*Steve Awodey*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0006
- Subject:
- Mathematics, Algebra

This chapter focuses on another elementary universal structure, which is also an example of a universal that is not a limit. This important structure is called an ‘exponential’ and it can be thought ...
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This chapter focuses on another elementary universal structure, which is also an example of a universal that is not a limit. This important structure is called an ‘exponential’ and it can be thought of as a categorical notion of a ‘function space’. Topics discussed include exponential in a category, Cartesian closed categories, Heyting algebras, and equational definition. Exercises are provided in the last part of the chapter.Less

This chapter focuses on another elementary universal structure, which is also an example of a universal that is not a limit. This important structure is called an ‘exponential’ and it can be thought of as a categorical notion of a ‘function space’. Topics discussed include exponential in a category, Cartesian closed categories, Heyting algebras, and equational definition. Exercises are provided in the last part of the chapter.

*Steve Awodey*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0009
- Subject:
- Mathematics, Algebra

This chapter focuses on notion of adjoint functor, which applies everything that has been learned so far to unify and subsume all the different universal mapping properties encountered, from free ...
More

This chapter focuses on notion of adjoint functor, which applies everything that has been learned so far to unify and subsume all the different universal mapping properties encountered, from free groups to limits to exponentials. It also captures an important mathematical phenomenon that is invisible without the use of the lens of category theory. It is argued that adjointness is a concept of fundamental logical and mathematical importance not captured elsewhere in mathematics. Topics discussed include hom-set definition, examples of adjoints, order adjoints, quantifiers as adjoints, RAPL 197, locally cartesian closed categories, and the adjoint functor theorem.Less

This chapter focuses on notion of adjoint functor, which applies everything that has been learned so far to unify and subsume all the different universal mapping properties encountered, from free groups to limits to exponentials. It also captures an important mathematical phenomenon that is invisible without the use of the lens of category theory. It is argued that adjointness is a concept of fundamental logical and mathematical importance not captured elsewhere in mathematics. Topics discussed include hom-set definition, examples of adjoints, order adjoints, quantifiers as adjoints, RAPL 197, locally cartesian closed categories, and the adjoint functor theorem.