Walter P Tholen: 'What is monoidal topology?'

22nd of July, 2021. Part of the Topos Institute Colloquium.
-----
Abstract: Monoidal topology may be seen as being inspired by some visionary remarks in Hausdorff‘s „Grundzüge der Mengenlehre“ of 1914, and it takes its modern lead from two distinct seminal contributions of the early 1970s: the Manes-Barr presentation of topological spaces in terms of ultrafilter convergence axioms, and Lawvere’s presentation of metric spaces as small categories enriched in the extended non-negative half-line of the reals. Both types of spaces become instances of small so-called (T,V)-categories, where T is a Set-monad and V a (commutative) quantale, i.e. a small, thin and (symmetric) monoidal-closed category. The setting therefore allows for a general study of „spaces“ that combines geometric and numerical aspects in a natural way.

In this talk we present some key elements of the theory and its applications, showing in particular how the strictification and inversion of some naturally occurring inequalities in this  lax-monoidal setting leads to interesting topological properties and unexpected connections. Time permitting, we will also point to some on-going and future work in the area.