# Topology Seminar

**University of Aveiro, Department of Mathematics, December 14, 2018**

## Talks

**Sala Sousa Pinto**

**14:00-14:45** Adriana Balan (University Politehnica of Bucharest, Romania)

*Extending set-functors to generalised metric spaces*

**14:45-15:30** Isar Stubbe (Université du Littoral-Côte d’Opale, Calais, France.)

*Divisibility and Diagonals*

**15:30-16:00** Coffee break

**Sala 11.2.25**

**16:00-16:30** Willian Ribeiro (University of Coimbra)

*Generalized equilogical spaces*

**16:30-17:00** Pedro Nora (University of Aveiro)

*H + V and everything else*

**17:00-17:30** Renato Neves (Minho University)

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

## Abstracts

### Adriana Balan

*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.

### Isar Stubbe

*Divisibility and diagonals*

**Abstract.** (pdf)

### Willian Ribeiro

*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.

### Renato Neves

*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.