**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.