(Algebra, Logic and Topology Group)
14:00-14:30 Dirk Hofmann (CIDMA, Univ. Aveiro), Hausdorff and Vietoris combined
14:30-15:00 Willian Ribeiro (CMUC), Weak cartesian closedness in Top and (T,V)-Cat
15:00-16:00 break (Colloquium of the Department)
16:30-17:00 Renier Jansen (Univ. Free State, Bloemfontein, South Africa), Dual closure operators via constant subcategories
17:00-17:30 Eros Matrinelli (CIDMA, Univ. Aveiro), Injectives in the category of multiordered sets
17:30-18:00 Maria Manuel Clementino (CMUC), Lax orthogonal factorisations in (T,V)-categories
Lax orthogonal factorisations of (T,V)-categories
Maria Manuel Clementino
We introduce lax orthogonal factorisations in order-enriched categories and present several examples constructed using presheaf monads in the category of (T,V)-categories, for suitable monad T and quantale V.
Hausdorff and Vietoris combined
In this talk we present the construction of a monad which combines the Hausdorff metric and the Vietoris topology, on the category of metric compact spaces. We discuss some properties of this monad, and compare it with other "metric generalisations" of the Vietoris construction. This study also requires to transport some classical results of ordered topological spaces to metric topological spaces.
Dual closure operators via constant subcategories
We consider categories with an (E,M)-factorisation structure for sources or, equivalently, a cocomplete class E of morphisms. The classical adjunction between left and right constant subcategories is studied via a fixed reflective constant subcategory, i.e., a reflective subcategory which is closed under M-subobjects and E-images. Adjunctions between classes of subcategories are then sought to factorise the classical one via dual closure operators.
Injectives in the category of multiordered sets
In this seminar we will prove that the injective objects in the category of multiordered sets are quantales. This will be done by proving that quantales are algebras for a Kock-Zoberlein monad which resembles the power set monad on Set. Using this, we will give a new proof of a result first proven by Barr, namely that quantales are also injectives in the category of ordered monoids (with lax morphisms).
Time allowed, we will discuss the connection between Isbell duality and injective hulls.
Weak cartesian closedness in Top and (T,V)-Cat
In this talk we present the notion of weak cartesian closedness and sketch the construction of weak exponentials of topological spaces and, more generally, of (T,V)-categories. We relate this notion with the concept of cartesian pre-closedness, which is needed for the study of cartesian closedness of exact completions of categories.