From a Geometrical Point of View (eBook)

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.

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