arXiv subjects

Jorge Picado

Publications and source records attributed to Jorge Picado.

3 recordsLinked to original sources

arXiv2026

Basic zero-dimensional spaces: a unifying framework for continuity and openness

Following a suggestion in a paper by Ern\'e, Picado and Pultr, we establish the category of basic zero-dimensional spaces and their basic continuous maps. The objects are closure spaces with a distributive closure system containing a specified meet-base of complemented members that make the category a common generalization of those of closure spaces, zero-dimensional topological spaces, Heyting semilattices, and locales (frames). We treat images, preimages, continuity, openness and closedness of maps between basic zero-dimensional spaces. Among our main results, we have a useful description of preimages for basic continuous maps (analogous to the one for localic preimage maps as coframe homomorphisms), and a Joyal-Tierney type theorem for basic open maps. A novel aspect of the category of basic zero-dimensional spaces is a duality principle that, among other things, enables results for basic closed maps to be obtained for free from those for basic open maps.

math.GN↗

arXiv2026

Continuity and openness of maps on locales by way of Galois adjunctions

We study four adjoint situations in pointfree topology that interchange images and preimages with closure and interior operators and establish with them a number of characterisations for meet-preserving maps, localic maps, open maps (in a broad sense) and open localic maps between locales. The principal and most attractive feature of these adjunctions is that they are all concerned with elementary ideas and basic concepts of localic topology: the use of the concrete language of sublocales and its technique simplifies the reasoning. We then revisit open localic maps in detail and present a new proof of Joyal-Tierney open mapping theorem. We end with a study of the interchange laws between preimages/images and closure/interior operators, making clear the similarities and differences with the classical realm.

math.GN↗

arXiv2016

Tensor products and relation quantales

A classical tensor product $A \,\otimes\, B$ of complete lattices $A$ and $B$, consisting of all down-sets in $A \times B$ that are join-closed in either coordinate, is isomorphic to the complete lattice $Gal(A,B)$ of Galois maps from $A$ to $B$, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudo\-complementedness. We show that the truncated tensor product of a complete lattice $B$ with itself becomes a quantale with the closure of the relation product as multiplication iff $B$ is pseudocomplemented, and the tensor product has a unit element iff $B$ is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets.

math.CT↗