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MFPS 17 Preliminary Version A Relationship between Equilogical Spaces

Summary: MFPS 17 Preliminary Version
A Relationship between Equilogical Spaces
and Type Two E ectivity
Andrej Bauer 1
Institut Mittag-Leer
The Royal Swedish Academy of Sciences
In this paper I compare two well studied approaches to topological semantics|
the domain-theoretic approach, exempli ed by the category of countably based
equilogical spaces, Equ, and Type Two E ectivity, exempli ed by the category of
Baire space representations, Rep(B ). These two categories are both locally cartesian
closed extensions of countably based T 0 -spaces. A natural question to ask is how
they are related.
First, we show that Rep(B ) is equivalent to a full core ective subcategory of Equ,
consisting of the so-called 0-equilogical spaces. This establishes a pair of adjoint
functors between Rep(B ) and Equ. The inclusion Rep(B ) ! Equ and its core ection
have many desirable properties, but they do not preserve exponentials in general.
This means that the cartesian closed structures of Rep(B ) and Equ are essentially
di erent. However, in a second comparison we show that Rep(B ) and Equ do share a
common cartesian closed subcategory that contains all countably based T 0 -spaces.


Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University


Collections: Mathematics