Conceptions of Space and the Role of Axiomatization

William Lawvere
Wednesday, July 24, 2013 - from 09:30 to 10:30
The axioms in the Foundations of Algebraic Topology were not an idealist starting point but rather an extraction of structure inherent in previous practice, creatively made explicit to serve as a guide to enlightened further practice. This work with Steenrod was an important instance of Sammy’s progressive use of axiomatization as a tool for ongoing research. It deeply impressed the students at Columbia University in the work on the foundations of universal algebra, of set theory and logic. It strongly influenced the conceptual underpinning of the topos foundations of algebraic geometry and of smooth geometry, namely the determination of a space by its incidence relations between figures (which were whimsically called ‘points’ by some) as clearly expressed in the first paragraph of the 1950 paper of Eilenberg & Zilber that introduced Simplicial Sets. Sammy’s philosophy of mathematics is reflected in his successful collaborations, with Steenrod, with Zilber, with Mac Lane, with Cartan. This philosophy and these collaborations were part of the attraction that led to the remarkable concentration of students and young researchers at Columbia in 1960. Also in 1960, in Cartan’s Paris seminar where simplicial sets had been further studied, Grothendieck elaborated a categorical approach to Complex Analytic Geometry. The simplicial and analytic categories were two of the first toposes studied explicitly but, as categories of all spaces of a given kind, they are different from the localic sort associated to a given topological space. In our struggle today to clarify the nature of the several classes of toposes and their relationships, Sammy’s axiomatic method still provides fundamental guidelines. In particular, his hero Hurewicz had done crucial work in the 1940’s on map spaces, that Sammy axiomatized in his 1965 collaboration with Kelly (using the crucial tool made explicit by his student Kan in 1958). The results of that collaboration continue to play a foundational role in the study of toposes and their applications to functional analysis.