The category of equilogical spaces and the effective topos as homotopical quotients

We show that the two models of extensional type theory, those given by the category of equilogical spaces and by the effective topos, are homotopical quotients of categories of 2-groupoids

Quotient completion for the foundation of constructive mathematics

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.Comment: 32 page

Elementary quotient completion

We extend the notion of exact completion on a weakly lex category to elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which freely adds effective quotients and extensional equality. We note that the elementary quotient completion can be obtained as the composite of two free constructions: one adds effective quotients, and the other forces extensionality of maps. We also prove that each construction preserves comprehensions

Relational Parametricity for Computational Effects

According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Many programs, however, exhibit computational effects, and are not accounted for by the standard theory of relational parametricity. In this paper, we develop a foundational framework for extending the notion of relational parametricity to programming languages with effects.Comment: 31 pages, appears in Logical Methods in Computer Scienc

Preface Volume 29

AbstractThis volume contains the proceedings of the 8th conference on Category Theory and Computer Science (CTCS '99) which was held in Edinburgh, UK, from September 10 to September 12, 1999.The purpose of the conference series is the advancement of the foundations of computing using the tools of category theory. While the emphasis is upon applications of category theory, it is recognized that the area is highly interdisciplinary.Previous meetings have been held in Guildford (Surrey), Edinburgh, Manchester, Paris, Amsterdam, Cambridge, and S. Margherita Ligure (Genova).Out of 39 submissions the Programme Committe has selected 19 for presentation at the Conference. The programme also included invited talks by Ryu Hasegawa (Tokyo), Peter Freyd (Pennsylvania), Marcelo Fiore (Sussex), Doug Smith (Kestrel Institute).We wish to thank the anonymous referees and the Programme Committee members: Jiri Adamek, Nick Benton, Rick Blute, Thierry Coquand, Martin Escardo, Masahito Hasegawa, Martin Hofmann (Chair), Peter O'Hearn, Dusko Pavlovic, Horst Reichel, Giuseppe Rosolini, Andre Scedrov.We also wish to thank the ENTCS editor Michael Mislove for encouragement and technical support

Frames and Topological Algebras for a Double-Power Monad

We study the algebras for the double power monad on the Sierpinski space in the Cartesian closed category of equilogical spaces and produce a connection of the algebras with frames. The results hint at a possible synthetic, constructive approach to frames via algebras, in line with that considered in Abstract Stone Duality by Paul Taylor and others