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Elements of -Category Theory [Kietas viršelis]

(The Johns Hopkins University, Maryland), (Macquarie University, Sydney)
  • Formatas: Hardback, 770 pages, aukštis x plotis x storis: 234x155x45 mm, weight: 1210 g, Worked examples or Exercises
  • Serija: Cambridge Studies in Advanced Mathematics
  • Išleidimo metai: 10-Feb-2022
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108837980
  • ISBN-13: 9781108837989
Kitos knygos pagal šią temą:
Elements of -Category TheoryDidesnis paveikslėlis
  • Formatas: Hardback, 770 pages, aukštis x plotis x storis: 234x155x45 mm, weight: 1210 g, Worked examples or Exercises
  • Serija: Cambridge Studies in Advanced Mathematics
  • Išleidimo metai: 10-Feb-2022
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108837980
  • ISBN-13: 9781108837989
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781108837989.html
Raktažodžiai:
The language of 8-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an 8-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of 8-categories from first principles in a model-independent fashion using the axiomatic framework of an 8-cosmos, the universe in which 8-categories live as objects. An 8-cosmos is a fertile setting for the formal category theory of 8-categories, and in this way the foundational proofs in 8-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

The language of 8-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. This book develops a new, more accessible model-independent approach to the foundations of 8-category theory by studying the universe, or 8-cosmos, in which 8-categories live.

Recenzijos

'The book of Riehl and Verity is altogether a pedagogical introduction, a unified presentation and a foundation of higher category theory. The theory of -cosmoi is an elegant way of organising and developing the subject. The extension of category theory to -categories is by itself a miracle, vigorously presented in the book.' André Joyal, Université du Québec ą Montréal 'Emily and Dom have done what many thought impossible: they have written an introductory text on a model-independent approach to higher category theory. This self-contained text is ideal for both end-users and architects of higher category theory. Every page is bursting at the seams with gorgeous insights and the refreshingly candid delight the authors take in their subject.' Clark Barwick, University of Edinburgh 'This remarkable book starts with the premise that it should be possible to study -categories armed only with the tools of 2-category theory. It is the result of the authors' decade-long collaboration, and they have poured into it all their experience, technical brilliance, and expository skill. I'm sure I'll be turning to it for many years to come.' Steve Lack, Macquarie University

Daugiau informacijos

This book develops the theory of infinite-dimensional categories by studying the universe, or -cosmos, in which they live.
Preface xi
Aims xiii
Acknowledgments xvii
PART I BASIC ∞-CATEGORY THEORY
1(274)
1 ∞-Cosmoi and Their Homotopy 2-Categories
5(49)
1.1 Quasi-Categories
6(14)
1.2 ∞-Cosmoi
20(18)
1.3 Cosmological Functors
38(6)
1.4 The Homotopy 2-Category
44(10)
2 Adjunctions, Limits, and Colimits I
54(31)
2.1 Adjunctions and Equivalences
55(9)
2.2 Initial and Terminal Elements
64(4)
2.3 Limits and Colimits
68(10)
2.4 Preservation of Limits and Colimits
78(7)
3 Comma ∞-Categories
85(48)
3.1 Smothering Functors
87(5)
3.2 ∞-Categories of Arrows
92(6)
3.3 Pullbacks of Isofibrations
98(4)
3.4 The Comma Construction
102(7)
3.5 Representable Comma ∞-Categories
109(13)
3.6 Fibered Adjunctions and Fibered Equivalences
122(11)
4 Adjunctions, Limits, and Colimits II
133(41)
4.1 The Universal Property of Adjunctions
134(4)
4.2 ∞-Categories of Cones
138(5)
4.3 The Universal Property of Limits and Colimits
143(12)
4.4 Pointed and Stable ∞-Categories
155(19)
5 Fibrations and Yoneda's Lemma
174(63)
5.1 Cartesian Arrows
176(13)
5.2 Cartesian Fibrations
189(12)
5.3 Cartesian Functors
201(6)
5.4 Cocartesian Fibrations and Bifibrations
207(3)
5.5 Discrete Cartesian Fibrations
210(8)
5.6 The Representability of Cartesian Fibrations
218(4)
5.7 The Yoneda Lemma
222(13)
An Interlude On ∞-Cosmology
235(2)
6 Exotic ∞-Cosmoi
237(38)
6.1 The ∞-Cosmos of Isofibrations
238(5)
6.2 Flexible Weighted Limits
243(10)
6.3 Cosmologically Embedded ∞-Cosmoi
253(22)
PART II THE CALCULUS OF MODULES
275(104)
7 Two-Sided Fibrations and Modules
279(26)
7.1 Two-Sided Fibrations
280(11)
7.2 The co-Cosmos of Two-Sided Fibrations
291(5)
7.3 The Two-Sided Yoneda Lemma
296(3)
7.4 Modules as Discrete Two-Sided Fibrations
299(6)
8 The Calculus of Modules
305(36)
8.1 The Double Category of Two-Sided Isofibrations
307(8)
8.2 The Virtual Equipment of Modules
315(6)
8.3 Composition of Modules
321(10)
8.4 Representable Modules
331(10)
9 Formal ∞-Category Theory in a Virtual Equipment
341(38)
9.1 Liftings and Extensions of Modules
342(8)
9.2 Exact Squares
350(7)
9.3 Pointwise Right and Left Kan Extensions
357(4)
9.4 Formal Category Theory in a Virtual Equipment
361(11)
9.5 Weighted Limits and Colimits in ∞-Categories
372(7)
PART III MODEL INDEPENDENCE
379(122)
10 Change-of-Model Functors
383(38)
10.1 Cosmological Functors Revisited
384(5)
10.2 Cosmological Biequivalences
389(5)
10.3 Cosmological Biequivalences as Change-of-Model Functors
394(9)
10.4 Inverse Cosmological Biequivalences
403(18)
11 Model Independence
421(39)
11.1 A Biequi valence of Virtual Equipments
422(10)
11.2 First-Order Logic with Dependent Sorts
432(15)
11.3 A Language for Model Independent ∞-Category Theory
447(13)
12 Applications of Model Independence
460(41)
12.1 Opposite (∞, 1)-Categories and ∞-Groupoid Cores
461(9)
12.2 Pointwise Universal Properties
470(16)
12.3 Existence of Pointwise Kan Extensions
486(13)
Appendix of Abstract Nonsense
499(2)
Appendix A Basic Concepts of Enriched Category Theory
501(38)
A.1 Cartesian Closed Categories
503(3)
A.2 Enriched Categories and Enriched Functors
506(5)
A.3 The Enriched Yoneda Lemma
511(7)
A.4 Tensors and Cotensors
518(3)
A.5 Conical Limits and Colimits
521(3)
A.6 Weighted Limits and Colimits
524(8)
A.7 Change of Base
532(7)
Appendix B An Introduction to 2-Category Theory
539(28)
B.1 2-Categories and the Calculus of Pasting Diagrams
539(8)
B.2 The 3-Category of 2-Categories
547(2)
B.3 Adjunctions and Mates
549(5)
B.4 Right Adjoint Right Inverse Adjunctions
554(3)
B.5 Absolute Absolute Lifting Diagrams
557(3)
B.6 Representable Characterizations of 2-Categorical Notions
560(7)
Appendix C Abstract Homotopy Theory
567(52)
C.1 Abstract Homotopy Theory in a Category of Fibrant Objects
568(14)
C.2 Weak Factorization Systems
582(9)
C.3 Model Categories and Quillen Functors
591(8)
C.4 Reedy Categories as Cell Complexes
599(8)
C.5 The Reedy Model Structure
607(12)
APPENDIX OF CONCRETE CONSTRUCTIONS
619(2)
Appendix D The Combinatorics of (Marked) Simplicial Sets
621(62)
D.1 Complicial Sets
622(9)
D.2 The Join and Slice Constructions
631(11)
D.3 Leibniz Stability of Cartesian Products
642(11)
D.4 Isomorphisms in Naturally Marked Quasi-Categories
653(14)
D.5 Isofibrations between Quasi-Categories
667(10)
D.6 Equivalence of Slices and Cones
677(6)
Appendix E ∞-Cosmoi Found in Nature
683(26)
E.1 Quasi-Categorically Enriched Model Categories
684(7)
E.2 ∞-Cosmoi of (∞, 1)-Categories
691(8)
E.3 ∞-Cosmoi of (∞, n)-Categories
699(10)
Appendix F The Analytic Theory of Quasi-Categories
709(24)
F.1 Initial and Terminal Elements
709(4)
F.2 Limits and Colimits
713(1)
F.3 Right Adjoint Right Inverse Adjunctions
714(2)
F.4 Cartesian and Cocartesian Fibrations
716(9)
F.5 Adjunctions
725(8)
References 733(8)
Glossary of Notation 741(4)
Index 745
Emily Riehl is an associate professor of mathematics at Johns Hopkins University. She received her PhD from the University of Chicago and was a Benjamin Peirce and NSF postdoctoral fellow at Harvard University. She is the author of Categorical Homotopy Theory (Cambridge, 2014) and Category Theory in Context (2016), and a co-author of Fat Chance: Probability from 0 to 1 (Cambridge, 2019). She and her present co-author have published ten articles over the course of the past decade that develop the new mathematics appearing in this book. Dominic Verity is a professor of mathematics at Macquarie University in Sydney and is a director of the Centre of Australian Category Theory. While he is a leading proponent of 'Australian-style' higher category theory, he received his PhD from the University of Cambridge and migrated to Australia in the early 1990s. Over the years he has pursued a career that has spanned the academic and non-academic worlds, working at times as a computer programmer, quantitative analyst, and investment banker. He has also served as the Chair of the Academic Senate of Macquarie University, the principal academic governance and policy body.