**Equilogical spaces and filter spaces**
We present some comparisons between (apparently) different
cartesian closed extensions of the category of topological
spaces. Work in general topology produced interesting
quasitoposes of filter spaces where a notion of convergence
replaced that of neighbourhood system. Also, models of
mechanical computations consist of topological spaces
obtained from directed-complete partial orders endowed with the
topology of directed sups. These are cartesian closed, but need not be
closed under various useful constructions like subspaces or quotients
which are otherwise natural in the approach using (one of the various
notions of) filter spaces.
In an attempt to reunify the views, Dana Scott proposed the category
Equ of equilogical spaces. The construction has been compared to that
of the category Mod of the modest sets in the effective topos; in
fact, in both situations, one can explain local cartesian closure
based on the same general facts. And the crucial construction involved
is that of the exact completion of a category with finite limits.
We use these tools to compare equilogical spaces and filter spaces.

**Categorie d'Azumaya e gruppo di Brauer di una categoria monoidale chiusa**
Si mostra che le teorie classiche di Azumaya e Brauer sono sottoprodotti
della teoria della Cauchy completezza.

**Homotopy theoretic aspects of finite spaces**
The classical homotopy theory of compact polyhedra can be studied in the
context of finite T_0 spaces. In particular (at least in principle), the
vanishing of Whitehead products can be detected by machine computation. For
example, there exist finite models of the multiplications on the one, three
and seven spheres forming part of a (mostly undiscovered) finite
non-Hausdorff topological algebra.

**Lagrange-Good inversion from trace**
The Lagrange-Good inversion formula has been known for long
in mathematical analysis and enumerative combinatorics,
employed to compute the inverse of analytic functions
and to enumerate certain combinatorical structures. We give
a new proof of the Lagrange-Good inversion formula, using
several ideas hinted by results in theoretical computer science.
The sources of ideas are Girard's semantics of the lambda
calculus by analytic functors and Hasegawa's finding of
the relation between fixpoint and categorical trace by
Joyal, Street and Verity.