* Locales as "topology-free spaces"
@ 1997-02-21 16:27 categories
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From: categories @ 1997-02-21 16:27 UTC (permalink / raw)
To: categories
Date: Fri, 21 Feb 1997 11:34:36 +0000
From: Steve Vickers <sjv@doc.ic.ac.uk>
I am making two papers available as Departmental Research Reports:
"Topical Categories of Domains"
"Localic Completion of Quasimetric Spaces"
Both explore the idea of locales (and, indeed, toposes) as "topology-free
spaces". The technique is to work not with frames (point-free topologies)
but with presentations of them, understood as propositional geometric
theories whose models are the points. (But it is normally more convenient
to work with equivalent predicate theories.) Then -
* the geometric theory already determines an implicit topology on its models;
* any construction of models of one theory out of models of another
automatically determines a continuous map (or geometric morphism), just so
long as the construction is geometric.
In effect, a restriction to geometric mathematics removes the need to treat
topology explicitly, hence "topology-free spaces". Apparently, explicit
topology is needed to correct the over-credulousness of classical reasoning
principles, though in practice the geometric constraints often end up
forcing one to reintroduce the normal topological arguments in a different
guise.
The two papers test the applicability of the idea in the two areas of
domain theory and quasimetric spaces. Aside from the "topology-free space"
aspects, the papers develop some new results:
"Topical Categories of Domains" addresses categorical domain theory and
replaces the usual classes of objects and morphisms by toposes classifying
them. New general results concerning fixpoints of endo-geometric-morphisms
of toposes exploit their intrinsically topological nature to give a simple
approach to the solution of domain equations. The paper also gives a
summary of the constructive theory of Kuratowski finite sets and
establishes some limitations to the Cartesian closedness of Sets.
"Localic Completion of Quasimetric Spaces" proposes a construction of
locales in completion of quasimetric spaces (using ideas of flatness
deriving from Lawvere's enriched category account), studies the
powerlocales and shows that a limit map from a locale of Cauchy sequences
to the completion is triquotient in the sense of Plewe.
Paper copies are available from me; electronic copies are expected to be
available shortly in the Department of Computing's Research Report series
coordinated by Frank Kriwaczek (frk@doc.ic.ac.uk).
Steve Vickers.
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