Mathematical Conceptware: Category Theory: RALF KROMER. Tool and Object: A History and Philosophy of Category Theory

Abstract

Category theory has been used by mathematicians for more than sixty years now. Its importance in algebraic geometry, algebraic topology, and homological algebra is indisputable. Its overall significance in mathematics and the foundations of mathematics is still a matter of debate. An historical analysis of the origins and development of category theory (CT) might shed an interesting light on various issues related to this significance. Ralf Krömer’s book constitutes the first global attempt at such an analysis. It is based on extensive original sources, and it offers an original and challenging perspective. It definitely should be read by historians and philosophers of contemporary mathematics. The book’s main thesis is that a form of pragmatism best describes the implicit philosophy of mathematicians using and developing CT from 1945 to approximately 1970. The thesis is primarily defended by looking at the history of CT, and the latter is organized by following the interactions of CT with three fields: algebraic topology, homological algebra, and algebraic geometry. This does indeed roughly follow the actual development of the field and is thus entirely justified.