* A universal characterization of the unit interval
@ 2001-04-10 14:28 Martin Escardo
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From: Martin Escardo @ 2001-04-10 14:28 UTC (permalink / raw)
To: categories
The following extended abstract is now available on the Web.
A universal characterization of the closed Euclidean interval
ABSTRACT. We propose a notion of interval object in a category with
finite products, providing a universal property for closed and bounded
real line segments. The universal property gives rise to an analogue
of primitive recursion for defining computable functions on the
interval. We use this to define basic arithmetic operations and to
verify equations between them. We test the notion in categories of
interest. In the category of sets, any closed and bounded interval of
real numbers is an interval object. In the category of topological
spaces, the interval objects are closed and bounded intervals with the
Euclidean topology. We also prove that an interval object exists in
any elementary topos with natural numbers object.
http://www.cs.bham.ac.uk/~mhe/papers/lics2001-revised.ps
Best wishes,
Martin Escardo & Alex Simpson
--
P.s. A draft full version with proofs is available on-line at
http://www.dcs.ed.ac.uk/home/als/Research/interval.ps
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2001-04-10 14:28 A universal characterization of the unit interval Martin Escardo
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