Preprint: "A topos-theoretic approach to Stone-type dualities"

Preprint: "A topos-theoretic approach to Stone-type dualities"

[Olivia Caramello]

Dear All,

The following preprint is available from the Mathematics ArXiv at the
address http://arxiv.org/abs/1103.3493:

O. Caramello, "A topos-theoretic approach to Stone-type dualities"

Abstract:

We present an abstract unifying framework for interpreting Stone-type
dualities; several known dualities are seen to be instances of just one
topos-theoretic phenomenon, and new dualities are introduced. In fact,
infinitely many new dualities between preordered structures and locales or
topological spaces can be generated through our topos-theoretic machinery in
a uniform way. We then apply our topos-theoretic interpretation to obtain
results connecting properties of preorders and properties of the
corresponding locales or topological spaces, and we establish adjunctions
between various kinds of categories as natural applications of our general
methodology. In the last part of the paper, we exploit the theory developed
in the previous parts to obtain a topos-theoretic interpretation of the
problem of finding explicit descriptions of models of 'ordered algebraic
theories' presented by generators and relations, and give several examples
which illustrate the effectiveness of our methodology. In passing, we
provide a number of other applications of our theory to Algebra, Topology
and Logic.

This work represents a concrete implementation of the abstract methodologies
introduced in the paper "The unification of Mathematics via Topos Theory",
which I advertised on this list some months ago; incidentally, some
subscribers to this list might be interested in the Russian translation of
the latter paper by Yury Bratkov, now available from the ArXiv at the
address http://arxiv.org/abs/1104.0563.
   
Comments are welcome.

Best regards,
Olivia Caramello




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