Totally disconnected reflections Totally disconnected spaces (locales, toposes) have been defined by orthogonality to the unit interval. The totally disconnected reflection and associated factorization does not always exist, but it has been used in homotopy theory for locales and toposes (Joyal & Moerdijk 1990; Bunge 1992). More general settings have been studied by Hyland-Robinson & Rosolini (1987) and by Carboni & Pare (1993) and are relevant for the question of the existence of suitable factorizations involving totally disconnectecd morphisms of spaces. A similar study of objects orthogonal to the tiny object D in a model of Synthetic Differential Geometry (Freyd, Yetter, Lawvere, Bunge 1983) is useful in the subject.

Towards axiomatic domain theory, I: Categorical models of PML This is the first of two talks where we present a first investigation into axiomatic domain theory for the denotational semantics of pure functional programming languages: Plotkin's MetaLanguage PML (a type theory with sums, products, exponentials and recursive types based on the propositions-as-types paradigm) is given a denotational semantics defined on top of a categorical notion of model. Roughly, models of PML are

parameterised algebraically compact partial cartesian closed categories with coproducts.
The purpose of this talk is to state what this means and why it is suitable.

Rappresentazioni nei fasci per categorie di algebre finitamente presentate We present a conservative embedding of the opposite category of finitely presented Heyting algebras in a topos of sheaves in order to trasfer categorical properties of this, e.g. regularity. Then we can give a semantic proof of Pitts' theorem on definability of propositional quantifiers. We also study possible extensions to similar categories of modal algebras. The proof methods use games a la Ehrenfeucht. This is joint work with M. Zawadowski.