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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.
List of contents
Category Theory and Klein's Erlangen Program.- Introducing Categories, Functors and Natural Transformations.- Categories as Spaces, Functors as Transformations.- Discovering Fundamental Categorical Transformations: Adjoint Functors.- Adjoint Functors: What They are, What They Mean.- Invariants in Foundations: Algebraic Logic.- Invariants in Foundations: Geometric Logic.- Conclusion.
About the author
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.
Summary
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.
Additional text
From the reviews:
"The main and basic point is that category theory is thoroughly geometric. … The monograph is highly readable, and it should be valuable to every expert in category theory, every novice in category theory and every mathematician who is not interested at all in category theory." (Hirokazu Nishimura, Zentralblatt MATH, Vol. 1165, 2009)
“The creation of a book such as this marks an interesting point in the history and philosophy of category theory. … While reading the book, I used the opportunity to revisit several seminal papers and to marvel at the advances they represented. The author clearly studied each paper thoroughly, had private communications with some pioneers, and succeeded quite well in describing the mathematical community’s mind set at the time of those advances. … I conclude by recommending the book.” (R. H. Street, Mathematical Reviews, Issue 2012 d)
Report
From the reviews:
"The main and basic point is that category theory is thoroughly geometric. ... The monograph is highly readable, and it should be valuable to every expert in category theory, every novice in category theory and every mathematician who is not interested at all in category theory." (Hirokazu Nishimura, Zentralblatt MATH, Vol. 1165, 2009)
"The creation of a book such as this marks an interesting point in the history and philosophy of category theory. ... While reading the book, I used the opportunity to revisit several seminal papers and to marvel at the advances they represented. The author clearly studied each paper thoroughly, had private communications with some pioneers, and succeeded quite well in describing the mathematical community's mind set at the time of those advances. ... I conclude by recommending the book." (R. H. Street, Mathematical Reviews, Issue 2012 d)
