28 Sept 2004 4:00 - 5:30 Gavin Seal Lax algebras and topological-like structures Abstract: Following the description by Manes [1] of the category of compact Hausdorff spaces as the Eilenberg-Moore category for the ultrafilter monad, Barr [2] showed that by weakening the axioms for a monad and the subsequent algebras, the Eilenberg-Moore category could be seen to be isomorphic to the category of topological spaces. Metric spaces also benefitted from a similar treatment in Lawere's fundamental paper [3], but for the identity monad this time. In recent years, a unified setting emerged, which allowed for a description of these categories, along with the categories of approach and uniform spaces, as categories of so-called "lax algebras" (see [4]). In this talk, we will present this approach and explain how lax algebras may be related to closure spaces. [1] E.G. Manes, "A triple theoretic construction of compact algebras", Springer Lecture Notes in Math. 80 (1969) 91-118. [2] M. Barr, "Relational algebras", Springer Lecture Notes in Math. 137 (1970) 39-55. [3] F.W. Lawere, "Metric spaces, generalized logic, and closed categories", Rend. Sem. Mat. Fis. Milano 43 (1973) 135-166. [4] M.M. Clementino, D. Hofman and W. Tholen, "One setting for all: metric, topology, uniformity, approach structure", Appl. Cat. Struct. 12 (2004) 127-154.