Marquis From a Geometrical Point of View

A Study of the History and Philosophy of Category Theory
1. Auflage 2008
ISBN: 978-1-4020-9384-5
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 14, 310 Seiten

Reihe: Logic, Epistemology, and the Unity of Science

From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein's Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane's work in the early 1940's and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.

Jean-Pierre Marquis teaches logic, epistemology and philosophy of science at the Université de Montréal. He has published papers on category theory, categorical logic, general philosophy of mathematics and philosophy of science.

Autoren/Hrsg.

Marquis, Jean-Pierre

Inhaltsverzeichnis

1;Acknowledgements;7
2;Contents;8
3;Introduction;10
4;Category Theory and Klein’s Erlangen Program;18
4.1;1.1 Eilenberg and Mac Lane’s Claim;18
4.2;1.2 Klein’s Program: Basic Aspects;21
4.3;1.3 Logical Remarks;41
4.4;1.4 Main Ontological and Epistemological Consequences of Klein’s Program;43
4.5;1.5 Groups and Geometries: Formal Supervenience and Reduction;45
4.6;1.6 Summing Up;48
5;Introducing Categories, Functors and Natural Transformations;50
5.1;2.1 From a Transformation Group to the Algebra of Mappings;53
5.2;2.2 Foundations of Category Theory;60
5.3;2.3 Philosophical Interlude: An Argument Against the Foundational Status of Category Theory;63
5.4;2.4 At Last, Natural Transformations;69
5.5;2.5 Extending Klein’s Program in the Wrong Direction;73
5.6;2.6 Category Theory: The First Phase 1945–1958;76
6;Categories as Spaces, Functors as Transformations;82
6.1;3.1 Universal Morphisms;83
6.2;3.2 Grothendieck and Abelian Categories;99
7;Discovering Fundamental Categorical Transformations: Adjoint Functors;118
7.1;4.1 The Background: Homotopy Theory and Category Theory;123
7.2;4.2 Kan’s Discovery;134
7.3;4.3 Kan’s 1958 Papers “Adjoint Functors”;141
8;Adjoint Functors: What They are, What They Mean;155
8.1;5.1 Adjointness;156
8.2;5.2 Equivalence of Categories Again;169
8.3;5.3 Back to Klein;172
8.4;5.4 From Groups to Groupoids;174
8.5;5.5 The Foundations of Category Theory. . . Again;183
9;Invariants in Foundations: Algebraic Logic;199
9.1;6.1 Lawvere’s Thesis;202
9.2;6.2 The Category of Categories as a Foundational Framework;205
9.3;6.3 The Elementary Theory of the Category of Sets;216
9.4;6.4 Categorical Logic: the Program;218
9.5;6.5 An Adjoint Presentation of Propositional Logic;224
9.6;6.6 Quantifiers as Adjoint Functors;228
9.7;6.7 Graphical Syntax: Sketches;233
9.8;6.8 Categorical Theories: Conceptual and Generic Structures;242
9.9;6.9 Summing Up;254
10;Invariants in Foundations: Geometric Logic;255
10.1;7.1 Grothendieck Toposes: Generalized Spaces;256
10.2;7.2 Elementary Toposes;269
10.3;7.3 Invariants Under Geometric Transformations;275
10.4;7.4 Invariants Under Logical Transformations;279
10.5;7.5 Invariant Foundational Frameworks;284
10.6;7.6 Using Geometric and Logical Invariants;290
10.7;7.7 Summing Up;291
11;Conclusion;293
12;References;299
13;Index;310