Preprint: "Ionads"

[Richard Garner <rhgg2@cam.ac.uk>]

Dear all,

At CT09, I gave a talk on what I then was calling
"generalised topological spaces", and now have named
"ionads". A few people asked if I had anything written down
concerning the content of that talk; this is to say that I
now do. A preprint is available at

http://arxiv.org/abs/0912.1415

An abstract follows.

Richard

---

The notion of Grothendieck topos may be considered as a
generalisation of that of topological space, one in which the
points of the space may have non-trivial automorphisms.
However, the analogy is not precise, since in a topological
space, it is the points which have conceptual priority over
the open sets, whereas in a topos it is the other way around.
Hence a topos is more correctly regarded as a generalised
locale, than as a generalised space. In this note, we
introduce a new notion---that of ionad---which stands in the
same relationship to a topological space as a (Grothendieck)
topos does to a locale. Some basic aspects of the theory are
developed and applications to topology, logic and geometry
are discussed.



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