Algebra and Logic Seminar - 3/12/2014

Wednesday, 3 December 2014, 3:15 p.m.

Lecturer: Dirk Hofmann (CIDMA, Departamento de Matemática, Universidade de Aveiro)

Title: "Spaces as enriched categories"

Local: Room 1.13, Edifício VII
Faculdade de Ciências e Tecnologia, Quinta da Torre, Caparica

Abstract: The topic of this talk has its roots in [3] where Lawvere makes the point that "... the kinds of structures which actually arise in the practice of geometry and analysis are far from being 'arbitrary' ..., as concentrated in the thesis that fundamental structures are themselves categories". While it is trivial to regard an individual group or an individual ordered set as a category, Lawvere made the fundamental observation that, within a natural generalisation of the definition of category, one also obtains metric spaces. Similarly, our work with topological spaces presented as convergence structures shaped the idea that topological spaces are categories, and therefore can be studied using notions and techniques from (enriched) Category Theory. With this motivation, we will present in this talk the concept of a topological theory [1] entailing a monad and a quantale as a possible "syntax" for Topology. Within this setting, our study of spaces is primary based on concepts and results like distributor, colimit, adjunction, dual space and the Yoneda lemma; however, we stress that this path leads us naturally to more traditional topics such as compact, locally compact and stably compact spaces, to the filter space and the Vietories construction, and to sober spaces and continuous lattices.
This talk is based on joint work with Maria Manuel Clementino (University of Coimbra), Walter Tholen (York University, Toronto) and Gavin Seal (École Polytechnique Fédérale de Lausanne) (see also [2]).