From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory

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Springer Science & Business Media, Nov 20, 2008 - Science - 310 pages
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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.

 

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This is a brilliant book. It explains the unexpected confluences and unifications of category theory as an extension of the Kleinian Erlangen program. It feels like the application of group transformations that "creates" space is simply setting out specific conditions generally without exception yields space. This to me is not merely an implication of theory to all potential practices, but the implicative order generated by the constraints of the transformations. How this avoids tautology and moves swiftly to show that various "complete" views are interchangeable, making them all party to a type of trivial pursuit makes me wonder whether it might not be better for the adoption of category theory into other fields of knowledge that the ultimate definition of morphism should not be changed to a chimera morphism. See, Efferman's Adjoint Functor paper. A chimera morphism at the very foundation of the theory then would have the conceptual seed to graph many analogies to ordinary and extraordinary phenomena. It would also make it easier, more convenient to accept that an asymmetric structure--the arrow--and its impetuous--is really about both "identity" [A is A] and difference [A is not-A]. Leibniz presumed such as core to his mathesis universalis. This intuition is captured and subsumed by the more general and inclusive f: A-> B.  

Contents

Category Theory and Kleins Erlangen Program
9
Basic Aspects
12
Encoding Basic Geometric Facts
15
The Irrelevance of the Nature of the Elements of a Space
25
124 Why a Transformation Group is not Quite Enough
28
125 But Then Again why a Group is Enough
29
126 Classifying Geometries
31
13 Logical Remarks
32
Adjoint Functors What They are What They Mean
147
51 Adjointness
148
52 Equivalence of Categories Again
161
53 Back to Klein
164
54 From Groups to Groupoids
166
55 The Foundations of Category Theory Again
175
Invariants in Foundations Algebraic Logic
191
61 Lawveres Thesis
194

14 Main Ontological and Epistemological Consequences of Kleins Program
34
Formal Supervenience and Reduction
36
16 Summing Up
39
Introducing Categories Functors and Natural Transformations
41
21 From a Transformation Group to the Algebra of Mappings
44
22 Foundations of Category Theory
51
An Argument Against the Foundational Status of Category Theory
54
24 At Last Natural Transformations
60
25 Extending Kleins Program in the Wrong Direction
64
The First Phase 19451958
67
Categories as Spaces Functors as Transformations
73
31 Universal Morphisms
74
Doing Duality without Elements
77
312 Universal Morphisms
86
32 Grothendieck and Abelian Categories
90
321 Abelian Categories
92
322 Representable Functors
102
Discovering Fundamental Categorical Transformations Adjoint Functors
109
Homotopy Theory and Category Theory
114
42 Kans Discovery
125
43 Kans 1958 Papers Adjoint Functors
132
62 The Category of Categories as a Foundational Framework
197
63 The Elementary Theory of the Category of Sets
208
the Program
210
65 An Adjoint Presentation of Propositional Logic
216
66 Quantifiers as Adjoint Functors
220
Sketches
225
Conceptual and Generic Structures
234
69 Summing Up
246
Invariants in Foundations Geometric Logic
247
Generalized Spaces
248
72 Elementary Toposes
261
73 Invariants Under Geometric Transformations
267
74 Invariants Under Logical Transformations
271
75 Invariant Foundational Frameworks
276
76 Using Geometric and Logical Invariants
282
77 Summing Up
283
Conclusion
285
References
291
Index
303
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About the author (2008)

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.

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