Matrix calculus for metric, topological and approach spaces
by
Walter Tholen
York University
Coauthors: Maria Manuel Clementino (University of Coimbra)

We take Lawvere's presentation of a metric space as a V-category (where V is the closed interval from 0 to infinity) as the starting point to describe topological spaces and approach spaces in a similar fashion. This leads to the definition and study of so-called (T, V)-algebras where V is any symmetric closed-monoidal category and T is a monad on Set which allows for a suitable extension to the (bi)category Mat(V) of V- matrices. The structures mentioned by the title, and many others, fit easily into this setting. In this talk we will focus on the contavariant/covariant behaviour of Alg(T, V) in the variables T and V and thereby exhibit many canonical functors relating the concrete structures mentinoned by the title. The topological properties of Alg(T, V) will be illustrated further in the talk by Maria Manuel Clementino.

Date received: July 20, 2001