Handbook of categorical algebra. 1, Basic category theory / Francis Borceux.

Format:

Book

Author/Creator:

Borceux, Francis, 1948- author.

Series:

Encyclopedia of mathematics and its applications ; v. 50.

Encyclopedia of mathematics and its applications ; volume 50

Language:

English

Subjects (All):

Categories (Mathematics).

Physical Description:

1 online resource (xv, 345 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1994.

Language Note:

English

Summary:

A Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence, with the first being essentially self-contained, and are accessible to graduate students with a good background in mathematics. Volume 1, which is devoted to general concepts, can be used for advanced undergraduate courses on category theory. After introducing the terminology and proving the fundamental results concerning limits, adjoint functors and Kan extensions, the categories of fractions are studied in detail; special consideration is paid to the case of localizations. The remainder of the first volume studies various 'refinements' of the fundamental concepts of category and functor.

Contents:

Cover; Half-title; Title; Copyright; Dedication; Contents; Preface to volume 1; Introduction to this handbook; 1 The language of categories; 1.1 Logical foundations of the theory; 1.2 Categories and functors; 1.3 Natural transformations; 1.4 Contravariant functors; 1.5 Full and faithful functors; 1.6 Comma categories; 1.7 Monomorphisms; 1.8 Epimorphisms; 1.9 Isomorphisms; 1.10 The duality principle; 1.11 Exercises; 2 Limits; 2.1 Products; 2.2 Coproducts; 2.3 Initial and terminal objects; 2.4 Equalizers, coequalizers; 2.5 Pullbacks, pushouts; 2.6 Limits and colimits; 2.7 Complete categories

2.8 Existence theorem for limits2.9 Limit preserving functors; 2.10 Absolute colimits; 2.11 Final functors; 2.12 Interchange of limits; 2.13 Filtered colimits; 2.14 Universality of colimits; 2.15 Limits in categories of functors; 2.16 Limits in comma categories; 2.17 Exercises; 3 Adjoint functors; 3.1 Reflection along a functor; 3.2 Properties of adjoint functors; 3.3 The adjoint functor theorem; 3.4 Fully faithful adjoint functors; 3.5 Reflective subcategories; 3.6 Epireflective subcategories; 3.7 Kan extensions; 3.8 Tensor product of set-valued functors; 3.9 Exercises

4 Generators and projectives4.1 Well-powered categories; 4.2 Intersection and union; 4.3 Strong epimorphisms; 4.4 Epi-mono factorizations; 4.5 Generators; 4.6 Projectives; 4.7 Injective cogenerators; 4.8 Exercises; 5 Categories of fractions; 5.1 Graphs and path categories; 5.2 Calculus of fractions; 5.3 Reflective subcategories as categories of fractions; 5.4 The orthogonal subcategory problem; 5.5 Factorization systems; 5.6 The case of localizations; 5.7 Universal closure operations; 5.8 The calculus of bidense morphisms; 5.9 Exercises; 6 Flat functors and Cauchy completeness

6.1 Exact functors6.2 Left exact reflection of a functor; 6.3 Flat functors; 6.4 The relevance of regular cardinals; 6.5 The splitting of idempotents; 6.6 The more general adjoint functor theorem; 6.7 Exercises; 7 Bicategories and distributors; 7.1 2-categories; 7.2 2-functors and 2-natural transformations; 7.3 Modifications and n-categories; 7.4 2-limits and bilimits; 7.5 Lax functors and pseudo-functors; 7.6 Lax limits and pseudo-limits; 7.7 Bicategories; 7.8 Distributors; 7.9 Cauchy completeness versus distributors; 7.10 Exercises; 8 Internal category theory

8.1 Internal categories and functors8.2 Internal base-valued functors; 8.3 Internal limits and colimits; 8.4 Exercises; Bibliography; Index

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88196-5

0-511-95793-9

1-107-10295-2

1-107-09467-4

1-107-08849-6

0-511-52585-0

OCLC:

708565553