 
Summary: Equilogical Spaces
Andrej Bauer, 1 Lars Birkedal, 2 Dana S. Scott 3
School of Computer Science, Carnegie Mellon University
Abstract
It is well known that one can build models of full higherorder dependent type theory
(also called the calculus of constructions) using partial equivalence relations (PERs)
and assemblies over a partial combinatory algebra (PCA). But the idea of categories
of PERs and ERs (total equivalence relations) can be applied to other structures
as well. In particular, we can easily dene the category of ERs and equivalence
preserving continuous mappings over the standard category Top 0 of topological
T 0 spaces; we call these spaces (a topological space together with an ER) equilogical
spaces and the resulting category Equ. We show that this categoryin contradis
tinction to Top 0 is a cartesian closed category. The direct proof outlined here uses
the equivalence of the category Equ to the category PEqu of PERs over algebraic
lattices (a full subcategory of Top 0 that is well known to be cartesian closed from
domain theory). In another paper with Carboni and Rosolini (cited herein) a more
abstract categorical generalization shows why many such categories are cartesian
closed. The category Equ obviously contains Top 0 as a full subcategory, and it nat
urally contains many other well known subcategories. In particular, we show why, as
a consequence of work of Ershov, Berger, and others, the KleeneKreisel hierarchy
