Subject: categories: Preprint: Bipolar spaces

From: claudio pisani <[email-masked]>
Date: Tue, 13 Dec 2005 13:33:23 +0100 (CET)

Dear categorists,
some of you might be interested in the following
paper, now available on arXiv
(http://arxiv.org/abs/math.CT/0512194)
"Bipolar spaces"
Claudio Pisani

Abstract:
Some basic features of the simultaneous inclusion of
discrete fibrations and discrete opfibrations on a
category A in the category of categories over A are
studied; in particular, the reflections and the
coreflections of the latter in the former are
considered, along with a negation-complement operator
which, applied to a discrete fibration, gives a
functor with values in discrete opfibrations (and vice
versa) and which turns out to be classical, in that
the strong contraposition law holds.

Such an analysis is developed in an appropriate
conceptual frame that encompasses similar "bipolar"
situations and in which a key role is played by
"cofigures", that is components of products;
e.g. the classicity of the negation-complement operator
corresponds to the fact that discrete opfibrations (or
in general "closed parts") are properly analyzed by
cofigures with shape in discrete fibrations ("open
parts"), that is, that the latter are "coadequate" for
the former, and vice versa.

In this context, a very natural definition of "atom"
is proposed and it is shown that, in the above
situation, the category of atoms reflections is the
Cauchy completion of A.
-----------

Of course, any comment or criticism is welcome. In
particular, I would like to know which facts, apart
from the formalism, sound familiar (e.g., concerning
the coreflection in discrete fibrations).

Best regards

Claudio

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