Category theory and applications: a textbook for beginners
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2018]
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | x, 294 Seiten Illustrationen 24 cm |
| ISBN: | 9789813231061 |
Staff View
MARC
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| 245 | 1 | 0 | |a Category theory and applications |b a textbook for beginners |c Marco Grandis, Università di Genova, Italy |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2018] | |
| 300 | |a x, 294 Seiten |b Illustrationen |c 24 cm | ||
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| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Categories (Mathematics) | |
| 650 | 4 | |a Mathematical analysis | |
| 650 | 4 | |a Morphisms (Mathematics) | |
| 650 | 4 | |a Homology theory | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
A
Introduction page 1
0.1 Categories 1
0.2 Universal properties 2
0.3 Diagrams in a category 3
0.4 Functors 4
0.5 Natural transformations and adjunctions 5
0.6 A brief outline 6
0.7 Classes of categories 7
0.8 Our approach 7
0.9 Literature 8
0.10 Notation 9
0.11 Acknowledgements 9
1 Categories, functors and natural transformations 10
1.1 Categories 10
1.2 Monoids and preordered sets as categories 17
1.3 Monomorphisrns and epimorphisms 22
1.4 Functors 26
1.5 Natural transformations and equivalence of categories 33
1.6 Categories of functors, representable functors 41
1.7 A category of Galois connections 45
1.8 Elementary categories of relations and partial mappings 52
2 Limits and colimits 61
2.1 Basic limits and colirnits 61
2.2 General limits and completeness 68
2.3 Pullbacks and pushouts 75
2.4 Subobjects and quotients, regular and strong 80
2.5 Factorisation systems 84
2.6 Inductive limits and filtered colimits 89
IX
X
Contents
2.7 Universal arrows and free objects 95
2.8 *Other universals 98
3 Adjunctions and monads 103
3.1 Adjoint functors 103
3.2 Properties of adjunctions 110
3.3 Comma categories 115
3.4 Monoidal categories and exponentials 120
3.5 The Adjoint Functor Theorem 128
3.6 Monads and algebras 131
3.7 Exercises and complements on monads 139
3.8 *Idempotent monads and idempotent adjunctions 141
4 Applications in Algebra 149
4.1 Free algebraic structures 149
4.2 Regular and Barr-exact categories 154
4.3 Varieties of algebras 156
4.4 Limits and free algebras 162
4.5 Relations for regular categories 166
5 Applications in Topology and Algebraic Topology 172
5.1 Adjoints and limits in Topology 173
5.2 Cylinder, cocylinder and homotopies 181
5.3 Simplicial sets 190
5.4 Cubical sets 196
5.5 Hints at Directed Algebraic Topology 202
6 Applications in Homological Algebra 208
6.1 Kernels and cokernels 209
6.2 Categories of lattices in Homological Algebra 212
6.3 Puppe-exact categories 217
6.4 Additive and abelian categories 225
6.5 Projective p-exact categories and projective spaces 231
6.6 Relations and induction for p-exact categories 237
7 Hints at higher dimensional category theory 246
7.1 From categories to 2-categories and bicategories 247
7.2 Double categories 254
7.3 Enriched, internal and ordered categories 262
7.4 * Double categories of lattices in Homological Algebra 269
7.5 * Double categories of relations in Homological Algebra 273
7.6 ^Factorisation systems as pseudo algebras for 2-monads 278
References 281
Index 287
Category Theory
and
Applications
A Textbook for Beginners
Category Theory now permeates most of Mathematics, large parts of
theoretical Computer Science and parts of theoretical Physics. Its
unifying power brings together different branches, and leads to a deeper
understanding of their roots.
This book is addressed to students and researchers of these fields and
can be used as a text for a first course in Category Theory. It covers its
basic tools, like universal properties, limits, adjoint functors and monads.
These are presented in a concrete way, starting from examples and
exercises taken from elementary Algebra, Lattice Theory and Topology,
then developing the theory together with new exercises and applications.
Applications of Category Theory form a vast and differentiated domain.
This book wants to present the basic applications and a choice of more
advanced ones, based on the interests of the author. References are
given for applications in many other fields.
World Scientific
www.worldscientific.com
10737 he
ISBN 978-981 -3231-06-1
789813
231061
|
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| dewey-raw | 512/.6 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-07-10T08:09:11Z |
| institution | BVB |
| isbn | 9789813231061 |
| language | English |
| lccn | 017042912 |
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| physical | x, 294 Seiten Illustrationen 24 cm |
| publishDate | 2018 |
| publishDateSearch | 2018 |
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| publisher | World Scientific |
| record_format | marc |
| spelling | Grandis, Marco Verfasser (DE-588)139362908 aut Category theory and applications a textbook for beginners Marco Grandis, Università di Genova, Italy New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2018] x, 294 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Categories (Mathematics) Mathematical analysis Morphisms (Mathematics) Homology theory Kategorientheorie (DE-588)4120552-2 gnd rswk-swf Kategorientheorie (DE-588)4120552-2 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030508558&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030508558&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Grandis, Marco Category theory and applications a textbook for beginners Categories (Mathematics) Mathematical analysis Morphisms (Mathematics) Homology theory Kategorientheorie (DE-588)4120552-2 gnd |
| subject_GND | (DE-588)4120552-2 |
| title | Category theory and applications a textbook for beginners |
| title_auth | Category theory and applications a textbook for beginners |
| title_exact_search | Category theory and applications a textbook for beginners |
| title_full | Category theory and applications a textbook for beginners Marco Grandis, Università di Genova, Italy |
| title_fullStr | Category theory and applications a textbook for beginners Marco Grandis, Università di Genova, Italy |
| title_full_unstemmed | Category theory and applications a textbook for beginners Marco Grandis, Università di Genova, Italy |
| title_short | Category theory and applications |
| title_sort | category theory and applications a textbook for beginners |
| title_sub | a textbook for beginners |
| topic | Categories (Mathematics) Mathematical analysis Morphisms (Mathematics) Homology theory Kategorientheorie (DE-588)4120552-2 gnd |
| topic_facet | Categories (Mathematics) Mathematical analysis Morphisms (Mathematics) Homology theory Kategorientheorie |
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