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.