Historical Foundations of Category Theory: Concepts and Contexts

Historical Foundations of Category Theory: Concepts and Contexts

I will be teaching the course Historical Foundations of Category Theory: Concepts and Contexts in spring 2026 at the University of Zurich. The course will be given in English. 

• Lecture: Tuesdays, 10am–12am 

• Material: 

• “Tool and Object: A History and Philosophy of Category Theory”, R. Krömer, Birkhäuser Basel (2007) 

• “Categories for the Working Mathematician”, S. Mac Lane, Graduate Texts in Mathematics, Springer-Verlag (1978) 

• “Topology: A Categorical Approach”, T.-D. Bradeley, T. Bryson, J. Terilla, The MIT Press (2020)

• Description of the course: This course explores the historical and philosophical evolution of category theory, following its path from a technical tool in algebraic topology to a central conceptual framework of twentieth-century mathematics. We will examine how category theory emerged in the 1940s through the collaboration of Eilenberg and Mac Lane, how it evolved through its roles in homological algebra, algebraic geometry, and foundational studies, and how it gradually transformed from a language of expression into an autonomous theory and even a proposed foundation of mathematics. Special emphasis will be placed on the interaction between mathematical practice and epistemology, in particular how categorical concepts such as functor, natural transformation, adjunction, and topos arose from concrete problems yet reshaped the very understanding of mathematical objects and structures. We will also discuss the work of Grothendieck, Lawvere, and others who extended the categorical viewpoint into geometry, logic, and the philosophy of mathematics, questioning the traditional set-theoretic conception of foundations. Students will engage with both primary historical texts and philosophical analysis, combining historical research with reflective inquiry. Each participant will select a  topic for a research essay and presentation. The use of AI tools for research and idea development is encouraged under ethical guidance, and assignments will be prepared in LaTeX to ensure professional mathematical presentation. Outstanding essays may contribute to a future collective publication on the history of category theory. This course is intended for students in mathematics who wish to understand the conceptual transformations that shaped modern mathematical thought. It is particularly suitable for those pursuing the teaching diploma in mathematics, as well as for students interested in the philosophical and historical foundations of mathematical ideas.