Topology Seminar

Extending set-functors to generalised metric spaces

Abstract. For a commutative quantale V (a lattice of “truth values”), the category V-cat can be perceived as a category of generalised metric spaces and non-expanding maps. Any type constructor (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor on V-cat. The extension is based on an obvious functor Set -> V-cat that is dense. For example, if the construction in question is taking powersets and V is the two-element quantale, then V-cat is the category of posets and the corresponding extension takes convex subsets with the Egli-Milner order; similarly, if V the lattice of positive real numbers, one obtains closed subspaces with the Pompeiu-Hausdorff distance. The easiest way to obtain concrete computations considers endofunctors on sets which preserve weak pullbacks (a pleasant property from a coalgebraic point of view), their extension to V-cat being then computed using the relation lifting.

Divisibility and diagonals

Abstract. (pdf)

Generalized equilogical spaces

Abstract. Introduced by Dana Scott in the late 90’s [1,2], equilogical spaces providenot onlyan alternative treatment to the problem of non-cartesian closedness of Top, the category of topological spaces and continuous maps, but also grounds for type theory. The category Equ of equilogical spaces and their morphisms is (co)complete and (co-)well-powered, and it contains Top as a full subcategory; moreover, from its connection with exact and regular completions [3,4], Equ is proven to be a quasitopos.In this talk, using the unifying setting of (T,V)-spaces and (T,V)-continuous maps [5], we discuss the problem of carrying the concept of equilogical spaces into other categories that are topological over Set. We establish a number of conditions that are necessary to achieve the above mentioned properties of Equ. Some of these conditions were studied in a recent work [6], while the other results can be found in [7].

[1] Dana S. Scott. A new category? domains, spaces and equivalence relations, manuscript, 1996.

[2] Andrej Bauer, Lars Birkedal, and Dana S. Scott. Equilogical spaces. Theoret. Comput. Sci., 315(1):35–59, 2004.

[3] Lars Birkedal, Aurelio Carboni, Giuseppe Rosolini, and Dana S. Scott. Type theory via exact categories(extended abstract). In Thirteenth Annual IEEE Symposium on Logic in Computer Science (Indianapolis, IN, 1998), pages 188–198. IEEE Computer Soc., Los Alamitos, CA, 1998.

[4] Giuseppe Rosolini. Equilogical spaces and filter spaces, manuscript, 1998.

[5] Maria Manuel Clementino and Walter Tholen. Metric, topology and multicategory—a common approach. J. Pure Appl. Algebra, 179(1-2):13–47, 2003.

[6] Maria Manuel Clementino, Dirk Hofmann, and Willian Ribeiro. Cartesian closed exact completions in topology. Preprint 18-46, Dept. Mathematics, Univ. Coimbra, arXiv 1811.03993, 2018.

[7] Willian Ribeiro. On generalized equilogical spaces. Preprint 18-50, Dept. Mathematics, Univ. Coimbra, arXiv:1811.08240, 2018.

When differential equations and programming constructs meet each other: a monadic perspective

Abstract. The recently introduced notions of guarded traced (monoidal) category and guarded (pre-)iterative monad aim at unifying different instances of partial iteration whilst keeping in touch with the established theory of total iteration and preserving its merits. In this talk we use these notions and the corresponding stock of results to examine different types of iteration for hybrid computations. As a starting point we use an available notion of hybrid monad restricted to the category of sets, and modify it in order to obtain a suitable notion of guarded iteration with guardedness interpreted as progressiveness in time - we motivate this modification by our intention to capture Zeno behaviour in an arguably general and feasible way. We illustrate our results with a simple programming language for hybrid computations and interpret it over the developed semantic foundations.