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

Introduction
1
Category Theory and Kleins Erlangen Program
9
Introducing Categories Functors and Natural Transformations
41
Categories as Spaces Functors as Transformations
73
Adjoint
109
What They are What They Mean
147
Algebraic Logic
191
Geometric Logic
247
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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