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Oxford Quantum Talks Archive

Maybe locales are made out of points after all

Martin Escardo, University of Birmingham
Categories, Logic and Foundations of Physics VII, September 2010, University of Birmingham

Like topology in analysis, locale theory is about open sets, continuous functions, compact spaces, approximation and limit processes, and things like that. Both topology and locale theory start with opens. In topology, an open is made out of points, but in locale theory, a point is made out of opens. The localic view makes physical and computational sense: points are infinitely small (and carry an infinite amount of information), and hence are not directly observable, but each point is uniquely characterized by its (infinite) collection of observable properties. The opens are the observables, and locale theory takes the notion of observation as primitive, and all other notions, including that of point, as derived. (Moreover, some perfectly good spaces in locale theory have a rich supply of opens without allowing any point at all, but this is not what I will emphasize in my talk). Although the match of (physical or computational) reality with locale theory is arguably better than with topology, locale theory may be more mathematically demanding, or at least is certainly unfamiliar to most of us. In this talk I'll discuss how one can think of locales as if they were made out of points, like the spaces of classical analysis and geometry, trying to make them more familiar, manageable, and intuitive, without loss of rigour, so that we can reason and work with them efficiently.

[video] [streaming video]