William Lawvere

Wednesday, July 24, 2013 - from 09:30
to 10:30

Abstract:

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.